201 Terms and Definitions in Algebra for the Mathematics Engineering Board Exam

Introduction

If there is one subject that cuts across every engineering discipline in the Philippine licensure examinations, it is algebra. Whether you are reviewing for the ECE, EE, CE, ME, ChE, or any other PRC board exam, algebra is always there. It shows up in the pure mathematics portion, in engineering sciences, and even embedded inside problems that appear to be about something else entirely. A circuit analysis problem is really an algebra problem in disguise. A beam loading problem reduces to a system of equations. A chemical mixture problem is a linear equation waiting to be set up. Algebra is the language that runs underneath all of engineering mathematics, and fluency in it is not optional.

What makes algebra challenging for many reviewees is not the difficulty of any single concept but the sheer breadth of the subject. Algebra covers everything from the most basic properties of real numbers to the theory of polynomial equations, from logarithms and exponential functions to matrices and determinants, from arithmetic sequences to infinite series. The PRC board exam draws items from all of these areas, and the questions range from straightforward substitution problems to multi-step word problems that require careful setup before any computation can begin. Knowing your formulas is necessary, but it is not sufficient. You also need to know when to apply each formula and how to recognize the type of problem in front of you.

This glossary was built specifically for Filipino engineering reviewees preparing for the PRC licensure examinations. The 201 terms collected here represent the full working vocabulary of algebra as it is tested in Mathematics engineering board exams. Each definition is written not just to tell you what a term means but to give you the board exam context for why it matters. Some terms are foundational concepts you learned in high school. Others are more specialized topics that appear only occasionally but can be the difference between a passing and failing score when they do show up. All of them belong in your review toolkit.

Work through this glossary systematically. Do not just read the definitions passively. After reading each term, ask yourself whether you can work a sample problem using that concept. If you cannot, flag it and make it part of your focused review. The engineering board exam rewards depth of understanding, not just surface-level recognition of terms. By the time you finish this list, you should have a clear picture of where your algebra strengths are and exactly which areas need more practice before exam day.

The 201 Algebra Terms and Definitions

1. Absolute Value

The distance of a number from zero on the number line, always expressed as a non negative value. The absolute value of x is written as |x| and defined as x when x is greater than or equal to zero, and as −x when x is less than zero. On the board exam, absolute value appears in equations (|x − 3| = 5), inequalities (|2x + 1| < 7), and function definitions. Solving absolute value equations requires splitting into two cases and checking both solutions for validity.

2. Absolute Value Equation

An equation that contains an absolute value expression. To solve |f(x)| = c where c is positive, split into two cases: f(x) = c and f(x) = −c, then solve each. If c is zero, only one case applies. If c is negative, there is no solution because absolute value is never negative. Board exam problems involving absolute value equations often have two valid solutions, one solution, or no solution, and choosing the wrong approach leads to missing solutions.

3. Absolute Value Function

The function defined as f(x) = |x|, which returns the non negative value of x regardless of its sign. Its graph forms a V shape with vertex at the origin. Transformations of the form f(x) = a|x − h| + k shift the vertex to (h, k) and change the width and orientation. The absolute value function is piecewise: it equals x for x ≥ 0 and −x for x < 0. Board exam problems on absolute value functions test graphing, transformation recognition, and solving equations or inequalities involving absolute value.

4. Absolute Value Inequality

An inequality involving an absolute value expression. The two standard forms are |f(x)| < c (which gives a bounded solution: −c < f(x) < c) and |f(x)| > c (which gives an unbounded solution: f(x) < −c or f(x) > c). The direction of the inequality determines whether the solution is an intersection or a union of intervals. Mixing up these two cases is one of the most common errors on board exam absolute value problems.

5. Addition Property of Equality

A fundamental property stating that adding the same quantity to both sides of an equation preserves equality. If a = b, then a + c = b + c for any value c. This property, along with the multiplication property of equality, forms the basis for all algebraic equation solving procedures. While it seems trivial, explicitly invoking this property is the correct justification when simplifying equations in formal algebraic proofs.

6. Additive Identity

The number zero, because adding zero to any number leaves the number unchanged: a + 0 = a. The additive identity is one of the field axioms that defines the structure of the real number system. In matrix algebra, the zero matrix serves as the additive identity because adding it to any matrix of the same dimensions produces the same matrix unchanged.

7. Additive Inverse

The additive inverse of a number a is −a, because a + (−a) = 0, the additive identity. Every real number has a unique additive inverse. In matrix algebra, the additive inverse of a matrix A is the matrix −A obtained by negating every entry. Understanding the additive inverse is foundational for solving linear equations by moving terms from one side to the other.

8. Adjoint Matrix

The transpose of the cofactor matrix of a square matrix, also called the adjugate matrix. The adjoint of matrix A is denoted adj(A). It is used in the formula for the inverse of a matrix: A⁻¹ = adj(A)/det(A). The adjoint matrix is important in board exam problems that ask for matrix inverses without using row reduction, particularly for 2×2 and 3×3 matrices where the formula is manageable.

9. Age Problem

A classic category of word problem in the PRC engineering board exam where the relationships between the current ages, past ages, or future ages of two or more people are expressed as algebraic equations. The key to solving age problems is to define a single variable for the present age of one person, express all other ages in terms of that variable, and then write an equation based on the given relationship. Always verify your answer by checking that all stated conditions are satisfied.

10. Algebraic Expression

A combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, exponentiation) that represents a quantity. Unlike an equation, an algebraic expression does not contain an equals sign. Examples include 3x² − 2x + 5 and √(a² + b²). Simplifying, evaluating, and manipulating algebraic expressions is the most fundamental skill in all of algebra.

11. Arithmetic Mean

The sum of a set of numbers divided by the count of numbers in the set. For two numbers a and b, the arithmetic mean is (a + b)/2. In the context of arithmetic sequences, the arithmetic mean of two terms is the term that falls exactly between them. Board exam problems frequently ask for the arithmetic mean of several quantities or use the arithmetic mean as a condition relating terms in a sequence.

12. Arithmetic Progression

A sequence of numbers in which each term after the first is obtained by adding a constant value called the common difference to the preceding term. The general term (nth term) is given by aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. The sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ) or Sₙ = n/2 × (2a₁ + (n − 1)d). Arithmetic progressions are among the most frequently tested sequence topics in the PRC engineering board exam.

13. Arithmetic Series

The sum of the terms of an arithmetic progression. The formula for the sum of the first n terms is Sₙ = n(a₁ + aₙ)/2, which is derived from the observation that pairing the first and last terms, the second and second to last terms, and so on, each pair sums to the same value. Board exam problems on arithmetic series typically give a combination of the first term, last term, common difference, number of terms, and sum, and ask you to find the missing quantities.

14. Associative Property

A property of addition and multiplication stating that the grouping of numbers does not affect the result. For addition: (a + b) + c = a + (b + c). For multiplication: (a × b) × c = a × (b × c). Subtraction and division are not associative. The associative property is used when regrouping terms for convenience in simplification, particularly when working with long polynomial or matrix expressions.

15. Asymptote

A line that a curve approaches but never reaches as the variable approaches a particular value or infinity. In algebra, asymptotes appear in rational functions. A vertical asymptote occurs where the denominator equals zero and the numerator does not. A horizontal asymptote describes the behavior of the function as x approaches positive or negative infinity. Identifying asymptotes is a standard task in board exam problems involving rational functions and their graphs.

16. Augmented Matrix

A matrix formed by appending the constant column of a system of linear equations to the coefficient matrix, separated by a vertical line. The augmented matrix [A|b] represents the system Ax = b. Row reduction of the augmented matrix using Gaussian elimination or Gauss Jordan elimination is the standard method for solving systems of linear equations in engineering problems. The augmented matrix is the starting point for all matrix based solution methods.

17. Base (Exponential)

The number that is raised to a power in an exponential expression. In the expression aⁿ, a is the base and n is the exponent. The base determines the fundamental behavior of the exponential function. Common bases in engineering board exam problems are 2 (binary systems), 10 (common logarithms), and e (natural logarithms and growth or decay problems). Manipulating bases correctly is essential for solving exponential equations.

18. Base (Logarithm)

The number b in the expression logₓ(y) = x, meaning bˣ = y. The base must be positive and not equal to 1. The two standard bases in engineering are base 10 (common logarithm, written log) and base e (natural logarithm, written ln). Changing between bases is done using the change of base formula: logₐ(x) = log(x)/log(a). Board exam problems frequently require converting between different logarithmic bases.

19. Binomial

A polynomial expression consisting of exactly two terms. Examples include x + 3, 2a − 5b, and x² + y². Binomials are the subject of several important algebraic formulas, including the difference of squares (a² − b² = (a+b)(a−b)), the sum and difference of cubes, and the binomial theorem. Recognizing a binomial form quickly allows you to apply the correct factoring or expansion formula without guessing.

20. Binomial Coefficient

The coefficient C(n,r) = n!/(r!(n−r)!), read as “n choose r,” that appears in the binomial theorem expansion of (a + b)ⁿ. Binomial coefficients form Pascal’s triangle, where each entry is the sum of the two entries directly above it. On the board exam, computing binomial coefficients correctly and efficiently is critical for solving binomial expansion problems under time pressure. These are also the same values used in probability combinations problems.

21. Binomial Expansion

The process of expanding (a + b)ⁿ using the binomial theorem, resulting in n+1 terms. Each term has the form C(n,r)aⁿ⁻ʳbʳ for r = 0, 1, 2, …, n. The expansion is symmetric: the coefficients of the rth term from the beginning and the rth term from the end are equal. Common board exam tasks include finding the rth term, the middle term (when n is even, there is one middle term at r = n/2; when n is odd, there are two middle terms), and the term with a specific power of the variable.

22. Binomial Theorem

A formula that gives the expansion of (a + b)ⁿ for any positive integer n. The general term in the expansion is C(n,r) × aⁿ⁻ʳ × bʳ, where C(n,r) is the binomial coefficient n!/(r!(n−r)!). The binomial theorem is a heavily tested topic in Philippine engineering board exams. Common problems ask for a specific term in the expansion (the rth term), the middle term, or the term containing a specific power of the variable. The term number is r + 1, not r.

23. Boolean Algebra

A branch of algebra that deals with variables that take only two values, typically 0 and 1 (or true and false). Boolean algebra uses operations of AND (multiplication), OR (addition), and NOT (complement) rather than ordinary arithmetic. It is the mathematical foundation of digital logic, computer circuits, and switching networks. For ECE and CpE board examinees, Boolean algebra is a required topic in digital systems and logic design.

24. Ceiling Function

A function that maps any real number to the smallest integer greater than or equal to it, denoted ⌈x⌉. For example, ⌈3.2⌉ = 4, ⌈−1.7⌉ = −1, and ⌈5⌉ = 5. The ceiling function is the counterpart of the floor function. Both functions appear in problems involving integer constraints, digital systems, and computational engineering contexts. Distinguishing between floor and ceiling is a tested skill in some PRC board exam problems.

25. Change of Base Formula

The formula that converts a logarithm from one base to another: logₐ(x) = logₓ(b)/log(b) or equivalently ln(x)/ln(a). This formula is essential because most scientific calculators only compute base 10 and base e logarithms directly. On the board exam, the change of base formula is used whenever a logarithm with an unusual base appears and needs to be evaluated or simplified using standard functions.

26. Characteristic Equation

In the context of matrices, the characteristic equation of a square matrix A is det(A − λI) = 0, where λ is the eigenvalue and I is the identity matrix. Solving the characteristic equation gives the eigenvalues of the matrix. While eigenvalue problems are more common in advanced engineering courses, the characteristic equation appears in some PRC board exam problems involving differential equations and vibration analysis.

27. Clock Problem

A classic category of algebra word problem involving the positions of the hour and minute hands of a clock. The minute hand gains 5.5 degrees per minute on the hour hand (since the minute hand moves 360 degrees per hour and the hour hand moves 30 degrees per hour). Clock problems ask for the times when the hands are coincident, perpendicular, or opposite each other. These problems appear regularly in the engineering board exam and require setting up a simple linear equation.

28. Coefficient

The numerical factor multiplying a variable or group of variables in an algebraic term. In the term 7x²y, the coefficient is 7. The coefficient of a term in a polynomial expansion determines the magnitude of that term’s contribution. In the binomial theorem, the binomial coefficient is specifically the coefficient of each term in the expansion. Identifying coefficients correctly is fundamental to polynomial manipulation and the factor and remainder theorems.

29. Coefficient Matrix

The matrix formed from the coefficients of the variables in a system of linear equations, not including the constant terms. For the system 2x + 3y = 5 and x − y = 1, the coefficient matrix is [2, 3; 1, −1]. The determinant of the coefficient matrix determines whether the system has a unique solution (nonzero determinant), no solution, or infinitely many solutions (zero determinant). Cramer’s rule and matrix inversion methods use the coefficient matrix as the starting point.

30. Cofactor

The cofactor of an element aᵢⱼ in a square matrix is defined as Cᵢⱼ = (−1)ⁱ⁺ʲ × Mᵢⱼ, where Mᵢⱼ is the minor of that element. The sign factor (−1)ⁱ⁺ʲ creates a checkerboard pattern of plus and minus signs across the matrix. Cofactors are used in computing determinants by cofactor expansion and in constructing the cofactor matrix, which leads to the adjoint and ultimately the matrix inverse.

31. Combined Variation

A relationship where a variable varies both directly with some quantities and inversely with others simultaneously. For example, if z varies directly as x and inversely as y², then z = kx/y², where k is the constant of variation. Combined variation problems appear frequently in Philippine engineering board exams in the form of word problems where multiple variables influence each other. The key step is writing the correct variation equation and then solving for the constant k using the given values.

32. Common Difference

The constant value added to each term to get the next term in an arithmetic progression, denoted d. It is calculated as d = aₙ₊₁ − aₙ for any consecutive pair of terms. The common difference can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence). Identifying the common difference from a given arithmetic sequence is always the first step before applying any arithmetic progression formula.

33. Common Logarithm

A logarithm with base 10, written as log(x) or log₁₀(x). The common logarithm is the standard logarithm in Philippine engineering review materials and in most scientific and engineering references. log(10) = 1, log(100) = 2, log(0.001) = −3, and log(1) = 0. Properties of common logarithms follow directly from the general logarithm laws. Most basic scientific calculators compute common logarithms using the LOG button.

34. Common Ratio

The constant factor multiplied by each term to get the next term in a geometric progression, denoted r. It is calculated as r = aₙ₊₁/aₙ for any consecutive pair of terms. The common ratio determines the behavior of the sequence: if |r| > 1 the terms grow without bound, if |r| < 1 the terms approach zero, if r = 1 all terms are equal, and if r = −1 the terms alternate in sign. Finding the common ratio is the first step in solving any geometric progression or geometric series problem.

35. Commutative Property

A property of addition and multiplication stating that the order of the numbers does not affect the result. For addition: a + b = b + a. For multiplication: ab = ba. Subtraction and division are not commutative, and matrix multiplication is also not commutative in general. The commutative property is used when rearranging terms in an expression for convenience during simplification.

36. Complete the Square

A technique for rewriting a quadratic expression ax² + bx + c into the form a(x − h)² + k by adding and subtracting a carefully chosen constant. This is done by taking half the coefficient of x, squaring it, and adding and subtracting it from the expression. Completing the square is used to derive the quadratic formula, convert a quadratic to vertex form, and solve quadratic equations that do not factor easily. It is also used in converting conic section equations to standard form in analytic geometry.

37. Complex Conjugate

The complex conjugate of a complex number a + bi is a − bi. The product of a complex number and its conjugate is always a real number: (a + bi)(a − bi) = a² + b². The conjugate is used to rationalize complex denominators when dividing complex numbers. On the board exam, the complex conjugate also appears in the theorem that complex roots of polynomials with real coefficients always come in conjugate pairs.

38. Complex Number

A number of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i² = −1. The real part is a and the imaginary part is b. Complex numbers extend the real number system and allow the square roots of negative numbers to be expressed. In engineering, complex numbers appear in AC circuit analysis (impedance), control systems (poles and zeros), and signal processing. The set of all complex numbers is denoted by ℂ.

39. Complex Roots

The roots of a polynomial equation that are complex numbers of the form a + bi where b ≠ 0. For polynomials with real coefficients, complex roots always appear in conjugate pairs: if a + bi is a root, then a − bi is also a root. This means polynomials of odd degree with real coefficients always have at least one real root. Board exam problems on theory of equations frequently test knowledge of this conjugate pair property.

40. Composite Function

A function formed by applying one function to the result of another, written as (f ∘ g)(x) = f(g(x)). The output of g becomes the input of f. The domain of the composite function is the set of all x values in the domain of g for which g(x) is in the domain of f. Board exam problems on composite functions typically ask for f(g(x)) or g(f(x)) given specific function definitions, and note that these two composites are generally not equal.

41. Compound Inequality

An inequality involving two separate inequality conditions connected by “and” (intersection) or “or” (union). A compound “and” inequality such as 2 < x < 7 requires x to satisfy both conditions simultaneously, giving a bounded interval. A compound “or” inequality such as x < 1 or x > 5 allows x to satisfy either condition, giving two separate unbounded intervals. Correctly identifying whether to use intersection or union is the key to solving compound inequalities on the board exam.

42. Conditional Equation

An equation that is true only for specific values of the variable, as opposed to an identity which is true for all permissible values. Most equations encountered in algebra are conditional equations. For example, 3x + 5 = 11 is a conditional equation true only when x = 2. Recognizing the difference between a conditional equation and an identity is important when verifying solutions or analyzing the nature of an equation.

43. Conjugate Pair Theorem

The theorem stating that if a polynomial has real coefficients and a + bi (with b ≠ 0) is a root, then its complex conjugate a − bi is also a root. This theorem is a direct consequence of the fact that coefficients are real, meaning the complex parts must cancel out when the polynomial is evaluated. On the board exam, this theorem is used to find additional roots when one complex root is given, or to verify that a polynomial has the correct number of real versus complex roots.

44. Constant

A fixed value that does not change within a given problem or expression. In the expression 5x² − 3x + 7, the number 7 is a constant. Constants are distinguished from variables, which can take on different values. In the context of variation problems, the constant of variation (often denoted k) relates the variables in a proportional relationship and must be determined from given data before the variation equation can be used.

45. Constant of Variation

The fixed proportionality constant k in a variation equation. For direct variation, y = kx. For inverse variation, y = k/x. For joint variation, z = kxy. For combined variation, z = kxⁿ/yᵐ. The constant of variation is always found first by substituting the given values of all variables into the variation equation and solving for k. Once k is known, the equation can be used to find any unknown value. Missing the step of solving for k first is a common board exam error.

46. Counting Numbers

Another name for natural numbers: 1, 2, 3, 4, 5, … These are the numbers used for counting objects. The counting numbers are the most basic number set in algebra, and all other number sets (integers, rationals, reals, complex numbers) are extensions built upon them. In combinatorics problems on the board exam, counting numbers represent the number of objects being arranged or selected.

47. Cramer’s Rule

A method for solving a system of n linear equations in n unknowns using determinants. Each unknown variable is expressed as the ratio of two determinants: the denominator is the determinant of the coefficient matrix, and the numerator is the determinant of the coefficient matrix with the column corresponding to the variable replaced by the constant column. Cramer’s rule is directly applicable when the coefficient matrix determinant is nonzero. It is a favorite method in Philippine engineering board exam problems because it gives a direct formula for each variable without requiring full row reduction.

48. Cubic Equation

A polynomial equation of degree 3, expressed in the general form ax³ + bx² + cx + d = 0 where a ≠ 0. A cubic equation always has exactly three roots (counting multiplicity) in the complex number system, and it always has at least one real root. Common board exam approaches for solving cubic equations include factoring by grouping, using the rational root theorem combined with synthetic division, and using the sum product relationships from Vieta’s formulas.

49. Degree of a Polynomial

The highest power of the variable in a polynomial expression. For example, the degree of 4x⁵ − 3x² + 7 is 5. The degree determines the maximum number of roots the polynomial can have, the end behavior of the polynomial function, and the number of terms in the binomial expansion. In the theory of equations, the degree is fundamental because it tells you how many roots to expect when solving the polynomial equation.

50. Dependent System

A system of linear equations that has infinitely many solutions. This occurs when the equations are not truly independent but are multiples of each other or linear combinations of each other, resulting in the same line (in 2D) or same plane (in 3D) being represented multiple times. In matrix form, a dependent system has a coefficient matrix with a determinant equal to zero and a consistent augmented matrix. The solution is expressed with one or more free variables.

51. Dependent Variable

The variable in a function or equation whose value is determined by the independent variable. In the function y = f(x), y is the dependent variable because its value depends on the chosen value of x. In word problems, the dependent variable is the quantity you are solving for or predicting, while the independent variable is the one you control or are given. Correctly identifying which variable is dependent is important for setting up variation and function problems.

52. Descartes’ Rule of Signs

A theorem that determines the maximum number of positive and negative real roots of a polynomial equation. The number of positive real roots equals the number of sign changes in the coefficients of f(x), or less than that by an even number. The number of negative real roots equals the number of sign changes in the coefficients of f(−x), or less than that by an even number. This rule does not give the exact number of roots but limits the possibilities, which is useful for narrowing down solutions on board exam multiple choice problems.

53. Determinant

A scalar value computed from the elements of a square matrix that encodes important properties of the matrix. For a 2×2 matrix with elements a, b, c, d, the determinant is ad − bc. For larger matrices, the determinant is computed by cofactor expansion along any row or column. A matrix with a nonzero determinant is invertible. The determinant also appears in Cramer’s rule and in the formula for the area of a triangle or parallelogram. Computing determinants accurately is a core skill for Philippine engineering board exams.

54. Difference of Cubes

The factoring identity a³ − b³ = (a − b)(a² + ab + b²). This is one of the standard special product formulas tested in Philippine engineering board exams. The factor (a − b) is the linear factor, and (a² + ab + b²) is the quadratic factor. Note that the quadratic factor does not factor further over the real numbers because its discriminant is negative. Recognizing a difference of cubes form quickly allows immediate factoring without trial and error.

55. Difference of Squares

The factoring identity a² − b² = (a + b)(a − b). This is one of the most frequently used factoring formulas in all of algebra and appears constantly in board exam problems. It applies whenever you have two perfect square terms being subtracted. Note that the sum of squares a² + b² does not factor over the real numbers. Recognizing the difference of squares pattern is an essential automatic skill for every engineering board examiner.

56. Digit Problem

A classic category of algebra word problem where the digits of a number are related by given conditions. The key setup is to express a two digit number as 10t + u (where t is the tens digit and u is the units digit) or a three digit number as 100h + 10t + u. When the digits are reversed, the new number is 10u + t. Board exam digit problems always provide two conditions, allowing you to set up two equations in two unknowns.

57. Diophantine Equation

An equation in which only integer solutions are sought, named after the ancient Greek mathematician Diophantus. A common example is finding integer values of x and y that satisfy ax + by = c. In Philippine engineering board exam problems, Diophantine style reasoning appears in word problems where the answer must be a whole number, such as the number of coins of each denomination, the number of items purchased, or the number of people in a group. Recognizing that a problem requires integer solutions narrows down the possible answers even when the algebra alone permits non integer results.

58. Direct Variation

A relationship between two variables where one variable is a constant multiple of the other. Written as y = kx, where k is the constant of variation. As x increases, y increases proportionally, and as x decreases, y decreases proportionally. The graph of a direct variation is a straight line through the origin. Direct variation is the simplest type of variation problem on the board exam and is often embedded within more complex combined variation problems.

59. Discriminant

The expression b² − 4ac found under the radical sign in the quadratic formula, denoted Δ (delta). The discriminant determines the nature and number of roots of a quadratic equation: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); if Δ < 0, there are two complex conjugate roots with no real roots. Board exam problems frequently ask for the nature of the roots of a quadratic equation, making the discriminant one of the most important concepts in all of algebra.

60. Distributive Property

The property stating that multiplication distributes over addition and subtraction: a(b + c) = ab + ac and a(b − c) = ab − ac. The distributive property is used in expanding products of polynomials, factoring expressions, and simplifying algebraic expressions. It is perhaps the most frequently applied algebraic property in all of engineering mathematics. The FOIL method for multiplying two binomials is a direct application of the distributive property applied twice.

61. Division Algorithm for Polynomials

The theorem stating that for any polynomial f(x) and any nonzero divisor polynomial d(x), there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that f(x) = d(x) × q(x) + r(x), where the degree of r(x) is less than the degree of d(x). This is the polynomial analog of integer division. The remainder theorem and synthetic division are both consequences of this algorithm and are tested regularly in Philippine engineering board exams.

62. Domain

The set of all permissible input values (x values) for a function. For a function to be defined at a particular x value, the output must exist as a real number. Common domain restrictions include: denominators cannot equal zero, expressions under square roots must be non negative, and logarithm arguments must be positive. Finding the domain of a given function is a standard task on board exams, particularly for rational functions, radical functions, and logarithmic functions.

63. Double Root

A root of a polynomial equation that occurs with multiplicity 2, meaning (x − r)² is a factor of the polynomial. At a double root, the graph of the polynomial touches the x axis but does not cross it. The discriminant of a quadratic equals zero when there is a double root. In the context of the binomial theorem or Vieta’s formulas, a double root counts twice in the sum and product of roots calculations.

64. Element (Matrix)

An individual entry in a matrix, identified by its row and column position. The element in the ith row and jth column of matrix A is denoted aᵢⱼ. When performing matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication, you operate on individual elements according to defined rules. Correctly identifying and referencing matrix elements is the starting point for all matrix computation problems on the board exam.

65. Elimination Method

A method for solving a system of linear equations by adding or subtracting multiples of the equations to eliminate one variable, reducing the system to a simpler form. Also called the addition method. For a 2×2 system, multiplying one or both equations by constants to make the coefficients of one variable equal and then subtracting eliminates that variable. The elimination method is particularly efficient for systems with two or three variables and is one of the most commonly tested solution methods in Philippine engineering board exams.

66. Empty Set

The set containing no elements, denoted by ∅ or {}. In the context of solving equations and inequalities, the empty set is the solution set when no value of the variable satisfies the given conditions. For example, the solution set of |x| = −3 is the empty set because absolute value is never negative. Recognizing when a problem has no solution and correctly stating that the solution set is empty prevents the board exam error of forcing a non existent solution.

67. Equation

A mathematical statement asserting that two expressions are equal, connected by an equals sign. An equation may be a conditional equation (true for specific values only), an identity (true for all permissible values), or a contradiction (true for no values). Solving an equation means finding all values of the variable that make both sides equal. The entire subject of algebra is built around the skill of setting up and solving equations that model real world and engineering problems.

68. Equivalent Equations

Two equations that have exactly the same solution set. Equivalent equations are produced by applying the properties of equality: adding the same quantity to both sides, multiplying both sides by the same nonzero quantity, or applying legitimate algebraic transformations. When solving equations, every valid step produces an equivalent equation. Introducing extraneous solutions (by squaring both sides, for example) breaks equivalence, which is why checking solutions in the original equation is always necessary.

69. Exponent

The number indicating how many times the base is multiplied by itself in an exponential expression. In aⁿ, n is the exponent. Exponents can be positive integers, negative integers, zero, fractions, or irrational numbers. The rules of exponents (product rule, quotient rule, power rule, zero exponent, negative exponent) are foundational algebra skills tested in every engineering board exam. Fractional exponents connect exponentiation to radicals: aᵐ/ⁿ = ⁿ√(aᵐ).

70. Exponential Equation

An equation in which the variable appears as an exponent. To solve exponential equations, either express both sides with the same base (if possible) and set exponents equal, or take the logarithm of both sides. For example, 2ˣ = 32 gives x = 5 by expressing 32 = 2⁵, while 3ˣ = 10 requires taking log of both sides to get x = log(10)/log(3). Exponential equations appear in board exam problems involving compound interest, population growth, radioactive decay, and capacitor charging or discharging.

71. Exponential Function

A function of the form f(x) = aˣ, where a is a positive constant not equal to 1. The natural exponential function f(x) = eˣ, where e ≈ 2.71828, is the most important exponential function in engineering. Exponential functions model growth and decay processes. They are always positive, have a horizontal asymptote at y = 0, and are either strictly increasing (when a > 1) or strictly decreasing (when 0 < a < 1). The domain is all real numbers and the range is all positive real numbers.

72. Extraneous Solution

A value that satisfies a transformed version of an equation but not the original equation. Extraneous solutions typically arise when both sides of an equation are squared (to eliminate a radical or absolute value), or when a rational equation is multiplied through by a variable expression. Always substitute all solutions back into the original equation to check for extraneous solutions. On the board exam, answer choices that include extraneous solutions are deliberately placed as distractors.

73. Factor

A number, variable, or expression that divides evenly into another expression. In the expression 6x²(x + 3), the factors are 6, x², and (x + 3). Factoring a polynomial means expressing it as a product of simpler polynomials or factors. The ability to factor efficiently is one of the most important computational skills in algebra, as it is used in solving polynomial equations, simplifying rational expressions, and finding roots of polynomials.

74. Factor Theorem

A special case of the remainder theorem stating that (x − r) is a factor of a polynomial f(x) if and only if f(r) = 0. In other words, r is a root of f(x) if and only if (x − r) divides f(x) with zero remainder. The factor theorem is used in combination with the rational root theorem to systematically find and verify factors of polynomials. On the Philippine engineering board exam, the factor theorem is one of the most directly tested theorems in the theory of equations.

75. Factoring by Grouping

A factoring technique used for polynomials with four or more terms where the terms are grouped in pairs (or other groupings), each pair is factored, and then a common binomial factor is extracted. For example, ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y). Factoring by grouping is the standard approach for cubic and quartic polynomials that do not factor by simpler methods and appears regularly in board exam polynomial problems.

76. Fibonacci Sequence

A sequence in which each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … The general rule is aₙ = aₙ₋₁ + aₙ₋₂. While the Fibonacci sequence is not as commonly tested as arithmetic and geometric progressions, it appears in some Philippine engineering board exam problems involving pattern recognition and sequences. The ratio of consecutive Fibonacci numbers approaches the golden ratio φ ≈ 1.618 as n increases.

77. Floor Function

A function that maps any real number to the largest integer less than or equal to it, denoted ⌊x⌋. For example, ⌊3.7⌋ = 3, ⌊−2.3⌋ = −3, and ⌊5⌋ = 5. The floor function is also called the greatest integer function. It appears in problems involving integer divisions, modular arithmetic, and digital system computations. Board exam problems involving the floor function require careful attention to negative numbers, where the result is less than (not closer to zero than) the input.

78. FOIL Method

A mnemonic for multiplying two binomials: First, Outer, Inner, Last. (a + b)(c + d) = ac + ad + bc + bd. The FOIL method is the direct application of the distributive property to binomial multiplication. While it applies specifically to two binomials, the principle extends to multiplying any polynomials by distributing each term of the first polynomial across all terms of the second. FOIL is one of the most basic and frequently used algebraic skills on the board exam.

79. Fraction

An expression of the form a/b where a is the numerator and b is the denominator and b ≠ 0. Fractions can be proper (|a| < |b|), improper (|a| ≥ |b|), or mixed numbers. Algebraic fractions (rational expressions) follow the same rules as numerical fractions for addition, subtraction, multiplication, and division. Simplifying fractions by canceling common factors and finding common denominators for addition are fundamental skills tested throughout the board exam.

80. Function

A rule that assigns exactly one output value to each input value. Formally, a function f from set A to set B assigns to each element in A exactly one element in B. A relation is a function if and only if no two ordered pairs share the same first element (the vertical line test on a graph). Functions are the central concept of all of engineering mathematics, and understanding domain, range, composition, inverse, and graph behavior is essential for success on the board exam.

81. Fundamental Theorem of Algebra

The theorem stating that every polynomial equation of degree n ≥ 1 with complex coefficients has exactly n roots in the complex number system, counting multiplicity. This theorem guarantees that a quadratic has exactly 2 roots, a cubic has exactly 3, and so on. It does not say all roots are real. Combined with the conjugate pair theorem, the fundamental theorem of algebra allows you to account for all roots of a polynomial and verify that you have found them all.

82. Geometric Mean

The nth root of the product of n numbers. For two positive numbers a and b, the geometric mean is √(ab). In a geometric sequence, the geometric mean of two terms is the term that falls exactly between them. The geometric mean is always less than or equal to the arithmetic mean (the AM GM inequality), with equality holding when all numbers are equal. Board exam problems on geometric mean often ask for the geometric mean between two given terms of a geometric sequence.

83. Geometric Progression

A sequence of numbers where each term after the first is obtained by multiplying the preceding term by a constant value called the common ratio r. The general term is aₙ = a₁ × rⁿ⁻¹. The sum of the first n terms is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. Geometric progressions appear in problems involving compound interest, population growth, and depreciation. They are among the most important sequences in the Philippine engineering board exam.

84. Geometric Series

The sum of the terms of a geometric progression. For a finite geometric series with first term a₁, common ratio r, and n terms, the sum is Sₙ = a₁(1 − rⁿ)/(1 − r). For an infinite geometric series with |r| < 1, the series converges to the sum S = a₁/(1 − r). The infinite geometric series formula is used in board exam problems involving repeating decimals, annuities, and converging processes. If |r| ≥ 1, the infinite series diverges and has no finite sum.

85. Greatest Common Factor (GCF)

The largest factor that divides each term of a polynomial or each number in a set without a remainder. Finding the GCF is always the first step in factoring a polynomial. For numbers, the GCF is found using prime factorization or the Euclidean algorithm. Extracting the GCF from a polynomial expression reduces it to a simpler form and is often the step that makes the remaining factoring straightforward. On the board exam, forgetting to factor out the GCF first leads to incomplete factoring.

86. Harmonic Mean

The reciprocal of the arithmetic mean of the reciprocals of a set of numbers. For two numbers a and b, the harmonic mean is 2ab/(a + b). In a harmonic sequence, consecutive terms have reciprocals forming an arithmetic progression. The harmonic mean appears in board exam problems involving average rates, parallel resistances, and combined work rates. For any two positive numbers, the harmonic mean is always less than or equal to the geometric mean, which is always less than or equal to the arithmetic mean.

87. Harmonic Progression

A sequence of numbers whose reciprocals form an arithmetic progression. For example, 1, 1/2, 1/3, 1/4, 1/5, … is a harmonic progression because the reciprocals 1, 2, 3, 4, 5, … form an arithmetic sequence. There is no simple general formula for the sum of a harmonic series. In Philippine engineering board exams, harmonic progression problems are typically solved by converting to the corresponding arithmetic progression using reciprocals and then applying arithmetic progression formulas.

88. Homogeneous System

A system of linear equations in which all constant terms are zero, expressed as Ax = 0. A homogeneous system always has at least one solution: the trivial solution x = 0 (all variables equal zero). If the determinant of the coefficient matrix is nonzero, the trivial solution is the only solution. If the determinant is zero, there are infinitely many non trivial solutions. In engineering, homogeneous systems appear in structural analysis, circuit theory, and eigenvalue problems.

89. Identity Matrix

A square matrix with 1s along the main diagonal and 0s everywhere else, denoted I or Iₙ for an n × n identity matrix. The identity matrix is the multiplicative identity for square matrices: AI = IA = A for any square matrix A of matching dimensions. The identity matrix plays the same role in matrix multiplication that the number 1 plays in ordinary arithmetic. Solving the matrix equation AX = B gives X = A⁻¹B only when A⁻¹ exists.

90. Imaginary Number

A number of the form bi where b is a real number and i is the imaginary unit with i² = −1. Imaginary numbers are the component added to real numbers to form complex numbers. Powers of i follow a repeating cycle of period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Board exam problems on imaginary numbers frequently ask for the value of iⁿ for a large n, which requires finding the remainder when n is divided by 4.

91. Inconsistent System

A system of linear equations that has no solution. This occurs when the equations represent parallel lines (in 2D) or parallel planes (in 3D) that never intersect. In matrix form, an inconsistent system is detected when row reduction produces a row of zeros in the coefficient matrix but a nonzero constant in the augmented column, giving a false statement like 0 = 5. A system is also inconsistent when the determinant of the coefficient matrix is zero and the system does not reduce to a dependent form.

92. Independent System

A system of linear equations that has exactly one unique solution. This occurs when the lines (in 2D) or planes (in 3D) intersect at a single point. In matrix form, an independent system has a nonzero coefficient matrix determinant, meaning the matrix is invertible. The unique solution can be found by any standard method: substitution, elimination, matrix inversion, or Cramer’s rule. Independent systems are the most common type tested in engineering board exam problems.

93. Independent Variable

The variable whose value is chosen freely in a function or equation. In the function y = f(x), x is the independent variable. In engineering problems, the independent variable is typically the quantity you are given or controlling, such as time, temperature, or input voltage. Identifying the independent variable helps in setting up the correct functional relationship between quantities in word problems and variation problems.

94. Index of a Radical

The number n in the expression ⁿ√x that indicates which root is being taken. For a square root, the index is 2 (usually omitted by convention). For a cube root, the index is 3. The index must be a positive integer. The index determines the domain: if n is even, the radicand must be non negative for real results; if n is odd, the radicand can be any real number. Converting between radical notation and fractional exponents (ⁿ√x = x^(1/n)) is a standard board exam skill.

95. Inequality

A mathematical statement that compares two expressions using symbols: < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, inequalities have solution sets consisting of ranges of values rather than discrete points. The properties of inequalities are similar to those of equations, except that multiplying or dividing both sides by a negative number reverses the direction of the inequality. This reversal rule is the most common source of errors in board exam inequality problems.

96. Infinite Geometric Series

A geometric series with infinitely many terms. It converges to a finite sum when the absolute value of the common ratio is less than 1: S∞ = a₁/(1 − r) where |r| < 1. If |r| ≥ 1, the series diverges and the sum is undefined. The infinite geometric series formula is directly applicable to problems involving repeating decimals (converting them to fractions) and to engineering problems involving damped oscillations and convergent processes.

97. Integer

A whole number that can be positive, negative, or zero: …, −3, −2, −1, 0, 1, 2, 3, … Integers form a subset of the rational numbers. The set of all integers is denoted ℤ. In engineering board exam problems, integers appear in number theory problems (divisibility, factors, multiples), sequence problems with integer terms, and in the rational root theorem where possible rational roots have integer numerators and denominators.

98. Interval Notation

A way of writing a set of real numbers using endpoints and brackets or parentheses. A square bracket [ or ] means the endpoint is included, while a parenthesis ( or ) means the endpoint is excluded. For example, [2, 5) represents all numbers from 2 to 5, including 2 but excluding 5. Interval notation is the standard way to express the solution sets of inequalities and the domains and ranges of functions. On the board exam, recognizing whether an endpoint should be open or closed depends on whether the original inequality is strict or non strict.

99. Inverse Function

A function f⁻¹ that reverses the action of the original function f. If f maps x to y, then f⁻¹ maps y back to x. A function has an inverse if and only if it is one to one (bijective). To find the inverse algebraically, swap x and y in the equation and solve for y. The graphs of f and f⁻¹ are reflections of each other across the line y = x. On the board exam, finding and verifying inverse functions is a tested skill, often connected to logarithm and exponential function pairs.

100. Inverse Matrix

For a square matrix A, the inverse matrix A⁻¹ is the matrix such that A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix. The inverse exists only if det(A) ≠ 0. For a 2×2 matrix with elements a, b, c, d, the inverse is (1/(ad−bc)) × [d, −b; −c, a]. The inverse matrix is used to solve systems of linear equations in matrix form: if Ax = b, then x = A⁻¹b. Computing the inverse of a 2×2 or 3×3 matrix is a standard board exam task.

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101. Inverse Variation

A relationship between two variables where their product is constant. Written as y = k/x or xy = k, where k is the constant of variation. As x increases, y decreases proportionally, and vice versa. The graph of an inverse variation is a hyperbola. Inverse variation problems appear frequently in Philippine engineering board exams, particularly in problems involving pressure and volume (Boyle’s Law), speed and time, and electrical resistance and current.

102. Investment Problem

A classic category of algebra word problem involving simple or compound interest. For simple interest: I = Prt, where P is the principal, r is the annual interest rate, and t is time in years. For compound interest: A = P(1 + r/n)^(nt), where n is the number of compounding periods per year. Investment problems on the engineering board exam typically involve finding the principal, interest rate, or time using these formulas, and sometimes involve two separate investments whose combined return satisfies a given condition.

103. Irrational Number

A real number that cannot be expressed as a ratio of two integers. Irrational numbers have non terminating, non repeating decimal expansions. Examples include √2, √3, π, and e. Irrational roots of polynomial equations with rational coefficients also occur in conjugate pairs (by the conjugate root theorem): if a + b√c is a root with a, b, c rational and √c irrational, then a − b√c is also a root.

104. Joint Variation

A relationship where one variable varies directly as the product of two or more other variables. Written as z = kxy, where k is the constant of variation and x and y are the independent variables. Joint variation problems are solved by first finding k from the given data, then using the variation equation to find the unknown quantity. On the board exam, joint variation appears in problems involving physical quantities that depend on multiple factors simultaneously.

105. Leading Coefficient

The coefficient of the term with the highest degree in a polynomial. In 5x³ − 2x + 7, the leading coefficient is 5. The leading coefficient determines the end behavior of the polynomial function (whether it rises or falls as x → ±∞) and appears in the denominator of the rational root candidates in the rational root theorem. Identifying the leading coefficient correctly is the starting step in applying Vieta’s formulas and the rational root theorem.

106. Least Common Multiple (LCM)

The smallest positive integer that is divisible by each of a given set of integers. The LCM of two numbers a and b is ab/GCF(a,b). For polynomials, the LCM is the polynomial of lowest degree divisible by each given polynomial. The LCM is used when finding common denominators for adding or subtracting rational expressions. Finding the LCM correctly before combining fractions is essential for solving rational equations on the board exam.

107. Like Terms

Terms in an algebraic expression that have the same variables raised to the same powers, differing only in their numerical coefficients. For example, 3x²y and −7x²y are like terms, but 3x²y and 3xy² are not. Like terms can be combined by adding or subtracting their coefficients while keeping the variable part unchanged. Combining like terms is one of the most basic and frequently applied simplification steps in algebra, and failing to identify like terms correctly is a common source of errors in expanding and simplifying expressions on the board exam.

108. Linear Equation

A polynomial equation of degree 1, expressed as ax + b = 0 where a ≠ 0. The solution is always x = −b/a, a single unique value. In two variables, a linear equation represents a straight line. In engineering board exam problems, linear equations are the most basic type and usually appear as part of systems of equations, word problems, or as simplified forms of more complex equations after algebraic manipulation.

109. Linear Inequality

An inequality involving a linear expression in one or more variables. In one variable, a linear inequality such as 3x − 2 > 7 is solved like a linear equation, with the exception that multiplying or dividing by a negative number reverses the inequality direction. The solution is an interval rather than a single value. In two variables, a linear inequality represents a half plane, which is relevant in linear programming problems appearing in operations research sections of engineering board exams.

110. Literal Equation

An equation containing two or more variables, where you solve for one variable in terms of the others. Examples include solving A = lw for l (giving l = A/w) or solving I = Prt for r (giving r = I/(Pt)). Literal equations appear in engineering board exam problems when you need to rearrange a formula to isolate a specific quantity. The solution process uses the same properties of equality as solving ordinary equations but produces a formula rather than a numerical answer.

111. Logarithm

The exponent to which a base must be raised to produce a given number. If bˣ = y, then logₓ(b) = x. Logarithms are the inverse of exponential functions. The three main properties of logarithms are: log(mn) = log m + log n, log(m/n) = log m − log n, and log(mⁿ) = n log m. These properties are used to simplify and solve logarithmic equations on the board exam. Understanding logarithms deeply is essential because they appear in engineering problems involving decibels, pH, Richter scale, and exponential decay.

112. Logarithmic Equation

An equation that contains a logarithm of an expression involving the variable. To solve logarithmic equations, use the properties of logarithms to combine log expressions into a single log, then convert to exponential form and solve. Always check that the solution makes the argument of every logarithm positive (domain check). Extraneous solutions from logarithmic equations are common on the board exam because negative arguments of logarithms are undefined.

113. Logarithmic Function

A function of the form f(x) = logₐ(x), where a > 0 and a ≠ 1. The domain is all positive real numbers and the range is all real numbers. The logarithmic function is the inverse of the exponential function. Its graph passes through (1, 0) and is increasing when a > 1 and decreasing when 0 < a < 1. The natural logarithmic function f(x) = ln(x) = logₑ(x) is the most important logarithmic function in engineering analysis and calculus applications.

114. Lower Bound (Roots)

A value L such that no real root of a polynomial is less than L. The lower bound theorem provides a criterion using synthetic division: if synthetic division by (x − c) with c < 0 produces alternating signs in the quotient row (including the remainder), then c is a lower bound for all real roots. The lower bound is used in combination with the upper bound to limit the range of real roots that need to be tested.

115. Matrix

A rectangular array of numbers arranged in rows and columns, enclosed in brackets. A matrix with m rows and n columns is called an m × n matrix. Matrices are used to represent systems of linear equations, transformations, networks, and data sets in engineering. The key matrix operations are addition, subtraction, scalar multiplication, matrix multiplication, transposition, and inversion. Matrix algebra is a major topic in Philippine engineering board exams, particularly for solving systems of equations.

116. Matrix Addition

The operation of adding two matrices of the same dimensions by adding corresponding elements. If A and B are both m × n matrices, then (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ. Matrix addition is commutative and associative. Matrix addition is undefined for matrices of different dimensions. On the board exam, matrix addition problems are straightforward once you verify that dimensions match.

117. Matrix Multiplication

The operation of multiplying two matrices where the number of columns in the first matrix equals the number of rows in the second matrix. If A is m × n and B is n × p, then AB is m × p with element (AB)ᵢⱼ = sum of aᵢₖ × bₖⱼ for k from 1 to n. Matrix multiplication is not commutative (AB ≠ BA in general) but is associative. The non commutativity of matrix multiplication is a classic point of confusion and a common source of errors in board exam matrix problems.

118. Mean Proportional

In a proportion of the form a/b = b/c, the term b is the mean proportional (or geometric mean) between a and c, satisfying b² = ac. The mean proportional appears in problems involving similar triangles, geometric sequences, and altitude relationships in right triangles, where the altitude to the hypotenuse is the mean proportional between the two segments it creates. On the board exam, recognizing a mean proportional setup allows a direct equation b² = ac instead of a longer derivation.

119. Minor (Matrix)

The determinant of the submatrix obtained by deleting the ith row and jth column of a square matrix, denoted Mᵢⱼ. The minor is used in the definition of the cofactor and in cofactor expansion for determinants. For a 3×3 matrix, each minor is the determinant of a 2×2 submatrix. Computing minors accurately is an important step in finding determinants by expansion and in computing the adjoint matrix for matrix inversion.

120. Mixture Problem

A classic category of algebra word problem where two or more substances with different concentrations or values are combined to produce a mixture with a specified concentration or value. The fundamental equation is: (amount₁)(concentration₁) + (amount₂)(concentration₂) = (total amount)(final concentration). Mixture problems appear in every Philippine engineering board exam and include alloy problems, solution problems, and price mixture problems. Drawing a table or diagram before setting up the equation prevents setup errors.

121. Modular Arithmetic

A system of arithmetic for integers where numbers wrap around after reaching a certain value called the modulus. The expression a mod m gives the remainder when a is divided by m. For example, 17 mod 5 = 2 because 17 = 3 × 5 + 2. Modular arithmetic appears in board exam problems involving clock problems (mod 12), calendar problems, cyclic patterns like powers of i (mod 4), and number theory problems involving remainders and divisibility.

122. Monomial

A polynomial consisting of exactly one term. Examples include 5, −3x², and 7a²b³. A monomial is the simplest type of polynomial. The degree of a monomial is the sum of the exponents of all its variables. Monomials are the building blocks of all polynomials, and operations on polynomials (addition, subtraction, multiplication) ultimately reduce to operations on monomials.

123. Motion Problem

A classic category of algebra word problem involving the relationship distance = rate × time (d = rt). Motion problems involve objects moving toward each other, away from each other, in the same direction, or in circular paths. The key is to identify what quantity each object contributes (distance, time, or rate) and to write an equation based on the geometric or physical relationship between the motions. Motion problems appear consistently in every Philippine engineering board exam.

124. Multiplicative Identity

The number 1, because multiplying any number by 1 leaves the number unchanged: a × 1 = a. In matrix algebra, the identity matrix I (with 1s on the main diagonal and 0s elsewhere) serves as the multiplicative identity because AI = IA = A for any square matrix A of compatible dimensions.

125. Multiplicative Inverse

The multiplicative inverse (or reciprocal) of a nonzero number a is 1/a, because a × (1/a) = 1. Every nonzero real number has a unique multiplicative inverse. In matrix algebra, the multiplicative inverse of an invertible matrix A is A⁻¹, satisfying AA⁻¹ = A⁻¹A = I. The existence of the multiplicative inverse is what allows division to be performed and equations to be solved by multiplying both sides by the inverse.

126. Multiplicity of a Root

The number of times a particular root appears as a solution of a polynomial equation, or equivalently the number of times the corresponding factor appears in the factored form. If (x − r)ᵏ divides f(x) but (x − r)ᵏ⁺¹ does not, then r has multiplicity k. A root with odd multiplicity crosses the x axis at that point, while a root with even multiplicity touches but does not cross the x axis. The sum of the multiplicities of all roots equals the degree of the polynomial.

127. Natural Logarithm

The logarithm with base e, where e ≈ 2.71828 is Euler’s number. Written as ln(x) or logₑ(x). The natural logarithm is the inverse of the natural exponential function eˣ. It is the preferred logarithm in calculus and in engineering applications involving growth, decay, and differential equations. The key properties are ln(eˣ) = x, e^(ln x) = x, and the same three product, quotient, and power rules that apply to all logarithms.

128. Natural Number

The counting numbers 1, 2, 3, 4, 5, … Some definitions include 0. Natural numbers are a subset of the integers, which are a subset of the rational numbers, which are a subset of the real numbers, which are a subset of the complex numbers. This hierarchy of number sets is fundamental to understanding the domain restrictions of algebraic expressions and functions tested on the board exam.

129. Negative Exponent Rule

The rule stating that a⁻ⁿ = 1/aⁿ for any nonzero base a. A negative exponent indicates a reciprocal. For example, x⁻³ = 1/x³ and 2⁻⁴ = 1/16. When simplifying expressions, negative exponents are usually converted to positive exponents by taking reciprocals. In scientific notation, negative exponents indicate values less than 1. The negative exponent rule is one of the fundamental laws of exponents and is used in virtually every problem involving rational or exponential expressions.

130. Number Line

A one dimensional representation of real numbers as points on a straight line, with a reference point at zero, positive numbers to the right, and negative numbers to the left. The number line is used to visualize solutions of inequalities, absolute value expressions, and interval notation. Understanding the number line is foundational for interpreting solution sets and checking whether a proposed solution is in the correct region.

131. Number Problem

A classic category of algebra word problem where relationships between unknown numbers are expressed as equations. Typical setups include: one number is twice another, the sum of two numbers is a given value, or the difference between two numbers satisfies a condition. Number problems are among the simplest word problem types on the Philippine engineering board exam but require careful translation of verbal conditions into algebraic equations without misinterpreting the language.

132. Numerical Coefficient

The constant factor in a term that contains variables. In the term −4x³y², the numerical coefficient is −4. In the expansion of (a + b)ⁿ, the numerical coefficients are the binomial coefficients. Correctly identifying and working with numerical coefficients is essential for polynomial operations, particularly when collecting like terms and applying the binomial theorem.

133. Oblique Asymptote

An asymptote that is neither horizontal nor vertical, occurring in rational functions where the degree of the numerator is exactly one more than the degree of the denominator. The oblique asymptote is found by performing polynomial long division and taking the quotient (ignoring the remainder). Oblique asymptotes describe the behavior of rational functions for large values of |x| and appear in advanced function analysis problems on the board exam.

134. One to One Function

A function in which each output value corresponds to exactly one input value. A function is one to one if and only if it passes the horizontal line test: no horizontal line intersects the graph more than once. One to one functions are the only functions that have inverses. The concept of one to one is important for determining whether a given function is invertible and is tested in board exam problems on inverse functions and function analysis.

135. Ordered Pair

A pair of numbers (x, y) where the order matters. The first element is the x coordinate and the second is the y coordinate. An ordered pair represents a point in the coordinate plane and is also used to represent a solution to a system of two equations in two unknowns. When solving a system by any method, the final answer is expressed as one or more ordered pairs, and failing to express it correctly in this form is a common board exam presentation error.

136. Partial Fraction Decomposition

The technique of expressing a rational expression as a sum of simpler fractions whose denominators are the factors of the original denominator. For example, (3x + 5)/((x+1)(x+2)) can be decomposed into A/(x+1) + B/(x+2). The method involves equating numerators after clearing denominators and solving for the unknown coefficients (A, B, etc.). Partial fraction decomposition is used in integral calculus and in inverse Laplace transforms, making it a critical algebra skill for all engineering disciplines.

137. Pascal’s Triangle

A triangular array of numbers where each entry is the sum of the two entries directly above it. The rows of Pascal’s triangle give the binomial coefficients for successive powers of (a + b): row 0 gives the coefficients for (a + b)⁰, row 1 for (a + b)¹, row 2 for (a + b)², and so on. Pascal’s triangle provides a quick visual method for finding binomial coefficients without computing factorials. It also has many other properties related to combinations, powers of 2, and the Fibonacci sequence.

138. Perfect Cube

A number or expression that is the cube of an integer or polynomial. For example, 8 = 2³, 27 = 3³, 125 = 5³, and x⁶ = (x²)³ are perfect cubes. Recognizing perfect cubes is the first step in applying the sum of cubes and difference of cubes factoring formulas. On the board exam, problems that ask you to factor expressions like 8x³ − 27 or 64 + a³b⁶ require identifying the perfect cube components before applying the standard formulas.

139. Perfect Square Trinomial

A trinomial that factors into the square of a binomial. The two standard forms are a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Recognizing a perfect square trinomial allows immediate factoring. On the board exam, perfect square trinomials appear in completing the square problems, in quadratic formula derivations, and in simplifying radical expressions. Checking whether the middle term is twice the product of the square roots of the first and last terms confirms the pattern.

140. Permutation

An arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is P(n,r) = n!/(n−r)!. When all n objects are used, the number of permutations is n!. Unlike combinations, permutations care about the order of selection. Permutation problems on the board exam include arrangements of letters in a word, seating arrangements, and ordered selections from a group. The distinction between “arrangement” (use permutation) and “selection” (use combination) is the key to choosing the correct formula.

141. Polynomial

An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀. Polynomials are classified by degree (linear, quadratic, cubic, quartic, quintic) and by the number of terms (monomial, binomial, trinomial). All polynomial operations (addition, subtraction, multiplication, division) are foundational skills for the engineering board exam.

142. Polynomial Equation

An equation of the form f(x) = 0 where f(x) is a polynomial. The roots (solutions) of a polynomial equation are the x values that make f(x) = 0. A polynomial of degree n has exactly n roots in the complex number system (by the fundamental theorem of algebra). The theory of equations, covering Vieta’s formulas, the rational root theorem, Descartes’ rule of signs, synthetic division, and the factor theorem, is the body of tools used to find and analyze these roots.

143. Polynomial Function

A function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the coefficients are constants and the exponents are non negative integers. Polynomial functions are continuous and smooth everywhere, with no breaks, holes, or asymptotes. Their graphs are classified by degree: linear functions are straight lines, quadratic functions are parabolas, and cubic functions have characteristic S shaped curves. On the board exam, distinguishing a polynomial function from a polynomial equation matters because the function describes a relationship between x and f(x) for all x, while the equation f(x) = 0 asks only for specific values of x where the function equals zero.

144. Power Rule (Exponents)

The rule stating that (aᵐ)ⁿ = aᵐⁿ when raising a power to another power. The exponents multiply. This is one of the key laws of exponents used in simplifying expressions involving nested exponents and in converting between radical and exponential forms. Combined with the product and quotient rules of exponents, the power rule handles virtually all exponential simplification encountered in Philippine engineering board exam problems.

145. Prime Number

A natural number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23. The number 1 is not prime by convention. The fundamental theorem of arithmetic states that every integer greater than 1 can be expressed uniquely as a product of prime numbers. Prime numbers appear in Philippine engineering board exam problems involving prime factorization, GCF, LCM, and number theory.

146. Product of Roots

By Vieta’s formulas, the product of all roots of the polynomial aₙxⁿ + … + a₀ = 0 equals (−1)ⁿ × a₀/aₙ. For a quadratic ax² + bx + c = 0, the product of the roots is c/a. For a cubic ax³ + bx² + cx + d = 0, the product of the roots is −d/a. Like the sum of roots formula, the product of roots formula allows you to find the product of all roots of a polynomial without solving for individual roots. This is a frequently tested shortcut in the theory of equations section of Philippine engineering board exams.

147. Product Rule (Exponents)

The rule stating that aᵐ × aⁿ = aᵐ⁺ⁿ when multiplying powers with the same base. The exponents add. This is one of the most basic and frequently used laws of exponents in algebra. The product rule applies only when the bases are the same. On the board exam, a common error is applying the product rule to expressions with different bases, so always verify that bases match before combining exponents.

148. Proportion

An equation stating that two ratios are equal: a/b = c/d. From a proportion, the cross multiplication property gives ad = bc. Proportions appear in direct variation, similar triangle problems, scale problems, and mixture problems. The mean extremes property (the product of the means equals the product of the extremes) is simply another way of stating cross multiplication and is the standard algebraic tool for solving proportion equations on the board exam.

149. Pure Imaginary Number

A complex number of the form bi where b is a nonzero real number and i is the imaginary unit. Pure imaginary numbers have no real part (a = 0). They appear as roots of equations like x² + 4 = 0, which gives x = ±2i. Pure imaginary roots of polynomials with real coefficients always come in conjugate pairs: if bi is a root, then −bi is also a root. This is a consequence of the complex conjugate root theorem applied to the case where the real part is zero.

150. Quadratic Equation

A polynomial equation of degree 2, expressed in the standard form ax² + bx + c = 0 where a ≠ 0. It can be solved by factoring, completing the square, or the quadratic formula. The quadratic formula is x = (−b ± √(b² − 4ac))/(2a). The nature of the roots is determined by the discriminant b² − 4ac. Quadratic equations are the most frequently solved equation type in all of algebra and appear in virtually every Philippine engineering board exam.

151. Quadratic Formula

The formula x = (−b ± √(b² − 4ac))/(2a) that gives the exact solutions of any quadratic equation ax² + bx + c = 0. It is derived by completing the square on the general form. The quadratic formula always works, even when factoring is difficult or impossible. On the board exam, the quadratic formula is the reliable fallback method when the equation does not factor neatly with integer coefficients. Memorizing this formula perfectly is non negotiable for every engineering board exam reviewer.

152. Quadratic Inequality

An inequality involving a quadratic expression, such as ax² + bx + c > 0. To solve a quadratic inequality, first find the roots of the corresponding quadratic equation, which divide the number line into intervals. Then test a value from each interval in the original inequality to determine which intervals satisfy the condition. The solution is typically expressed in interval notation. Quadratic inequalities appear on the board exam in problems involving the domain of functions with square roots, sign analysis, and optimization constraints.

153. Quartic Equation

A polynomial equation of degree 4, expressed in the general form ax⁴ + bx³ + cx² + dx + e = 0. A quartic equation has exactly four roots (counting multiplicity and complex roots) in the complex number system. Solution methods include factoring into two quadratics, using the rational root theorem combined with synthetic division, and using substitution for special forms like biquadratic equations (where only even powers of x appear). Biquadratic equations reduce to quadratics through the substitution u = x².

154. Quotient Rule (Exponents)

The rule stating that aᵐ/aⁿ = aᵐ⁻ⁿ when dividing powers with the same base. The exponent of the denominator is subtracted from the exponent of the numerator. When m < n, the result has a negative exponent, which indicates a reciprocal. When m = n, the result is a⁰ = 1. The quotient rule is used constantly in simplifying rational exponential expressions and in problems involving scientific notation in engineering computations.

155. Radical

An expression involving a root, written as ⁿ√x, where n is the index and x is the radicand. The most common radical is the square root (index 2), written as √x. Radicals are related to fractional exponents: ⁿ√x = x^(1/n). Simplifying radicals involves extracting perfect nth power factors from the radicand. Adding and subtracting radicals requires like radicals (same index and same radicand), similar to combining like terms in polynomial expressions.

156. Radical Equation

An equation that contains a radical expression with the variable inside the radical. To solve, isolate the radical on one side and raise both sides to the nth power to eliminate it. Because raising to even powers is not a reversible operation for all values, extraneous solutions may be introduced. Always substitute all solutions back into the original equation. On the board exam, radical equations usually have one valid solution and one extraneous solution, making the verification step essential.

157. Range (Function)

The set of all output values (y values) produced by a function for all inputs in its domain. Finding the range requires understanding how the function transforms the input. For a linear function, the range is all real numbers. For a quadratic function f(x) = a(x−h)² + k, the range is [k, ∞) when a > 0 or (−∞, k] when a < 0. The range of the common logarithm is all real numbers, while the range of the exponential function is all positive real numbers.

158. Ratio

The quotient of two quantities, expressing how many times one contains the other. A ratio a:b means a/b. Ratios are used throughout engineering in scaling, proportion problems, mixture problems, and geometric similarity. When working with ratios on the board exam, always ensure that both quantities being compared have the same units before computing the ratio. The simplest form of a ratio has no common factor between the two terms.

159. Rational Expression

An expression of the form p(x)/q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0. Rational expressions are the algebraic analog of fractions. Operations on rational expressions follow the same rules as operations on numerical fractions: multiply and divide by multiplying or dividing numerators and denominators, and add or subtract by finding a common denominator. Simplifying rational expressions by canceling common polynomial factors is a standard board exam skill.

160. Rational Number

A number that can be expressed as a ratio of two integers p/q where q ≠ 0. Rational numbers include all integers, all finite decimals, and all repeating decimals. The decimal expansion of a rational number either terminates or repeats. The set of all rational numbers is denoted ℚ. In the context of the rational root theorem, possible rational roots of a polynomial are ratios of factors of the constant term to factors of the leading coefficient.

161. Rational Root Theorem

A theorem that lists all possible rational roots of a polynomial equation with integer coefficients. If the polynomial is aₙxⁿ + … + a₀, then every rational root has the form ±p/q, where p is a factor of the constant term a₀ and q is a factor of the leading coefficient aₙ. The rational root theorem narrows down the candidates to test when solving polynomial equations by trial and division. Combined with synthetic division and the factor theorem, it is the standard method for solving polynomial equations on Philippine engineering board exams.

162. Rationalizing the Denominator

The process of eliminating radicals from the denominator of a fraction. For a monomial denominator like √3, multiply numerator and denominator by √3. For a binomial denominator like (2 + √5), multiply by the conjugate (2 − √5) to use the difference of squares identity. Rationalizing the denominator is a required simplification step in many board exam problems because standard answer formats do not have radicals in denominators. Forgetting to rationalize when the answer choices are all rationalized is a common source of missed matches.

163. Real Number

Any number that can be represented as a point on the number line. Real numbers include all rational numbers (integers, fractions, terminating and repeating decimals) and all irrational numbers (non terminating, non repeating decimals like √2 and π). The set of all real numbers is denoted ℝ. Real numbers form a complete ordered field, meaning they are ordered, closed under the four basic operations (with nonzero divisors), and have no gaps in the number line.

164. Reciprocal

The multiplicative inverse of a number or expression: the reciprocal of a is 1/a. The product of a number and its reciprocal is always 1. For a fraction a/b, the reciprocal is b/a. In solving equations, multiplying both sides by the reciprocal of a coefficient is the standard way to isolate a variable. Reciprocals also appear in harmonic progressions (where the reciprocals form an arithmetic progression) and in inverse variation (y = k/x).

165. Rectangular Matrix

A matrix that has a different number of rows and columns (m ≠ n). Rectangular matrices cannot have determinants, and the standard inverse does not apply to them. However, they can be multiplied (subject to dimension compatibility), transposed, and used in systems of equations that are overdetermined or underdetermined. The distinction between square and rectangular matrices determines which operations and properties apply.

166. Remainder Theorem

The theorem stating that when a polynomial f(x) is divided by (x − r), the remainder is f(r). In other words, you can find the remainder of polynomial division by simply evaluating the polynomial at x = r, without performing the full division. The remainder theorem is a powerful shortcut that appears frequently in Philippine engineering board exam problems asking for the remainder when a polynomial is divided by a linear factor. The factor theorem is a special case where f(r) = 0.

167. Root of a Polynomial

A value r such that f(r) = 0, where f(x) is a polynomial. Roots are also called zeros of the polynomial. The number of roots (counting multiplicity) equals the degree of the polynomial. Finding roots of polynomials is the central problem of the theory of equations and draws on the rational root theorem, factor theorem, remainder theorem, synthetic division, Vieta’s formulas, and Descartes’ rule of signs. All of these tools are tested in Philippine engineering board exams.

168. Scalar Multiplication (Matrix)

The operation of multiplying every element of a matrix by a constant (scalar). If k is a scalar and A is a matrix, then (kA)ᵢⱼ = k × aᵢⱼ. Scalar multiplication is commutative (kA = Ak), distributive over matrix addition (k(A + B) = kA + kB), and distributive over scalar addition ((k + m)A = kA + mA). On the board exam, scalar multiplication appears in problems involving matrix equations, determinant properties (factoring a scalar from a row multiplies the determinant by that scalar), and systems of equations.

169. Scientific Notation

A way of expressing very large or very small numbers in the form a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. For example, 0.00045 = 4.5 × 10⁻⁴ and 3,200,000 = 3.2 × 10⁶. Scientific notation uses the rules of exponents for multiplication and division. In engineering board exam problems involving calculations with very large or very small quantities (such as in electronics and electromagnetics), expressing results in scientific notation is often required.

170. Sequence

An ordered list of numbers following a defined pattern or rule. Each number in the sequence is called a term. Sequences can be finite (a limited number of terms) or infinite (continuing without end). The most important types of sequences in the Philippine engineering board exam are arithmetic progressions (constant difference), geometric progressions (constant ratio), and harmonic progressions (reciprocals form an arithmetic sequence). Recognizing the type of sequence from its terms is the first step in solving any sequence problem.

171. Series

The sum of the terms of a sequence. A finite series has a countable sum obtained by adding all terms. An infinite series may converge to a finite sum or diverge. The partial sum Sₙ is the sum of the first n terms. The formulas for arithmetic and geometric series are among the most frequently applied formulas in Philippine engineering board exams. Word problems involving total payments, accumulated savings, and depreciation over multiple periods are modeled by series.

172. Set

A collection of distinct objects called elements. Sets are denoted by listing elements in braces: {1, 2, 3}. Basic set operations include union (A ∪ B, all elements in A or B), intersection (A ∩ B, elements in both A and B), and complement (A’, elements not in A). Set theory provides the formal framework for discussing solution sets of equations and inequalities. In probability and counting problems on engineering board exams, Venn diagrams and set operations are essential tools.

173. Signum Function

A function that extracts the sign of a real number: sgn(x) = 1 if x > 0, sgn(x) = 0 if x = 0, and sgn(x) = −1 if x < 0. The signum function appears in problems involving piecewise defined functions and in digital signal processing contexts. While not as commonly tested as other functions in Philippine engineering board exams, recognizing and evaluating the signum function is a useful skill for advanced mathematics problems.

174. Simultaneous Equations

A system of two or more equations that must all be satisfied by the same set of variable values. Simultaneous equations are solved by substitution, elimination, matrix methods (Cramer’s rule, Gaussian elimination, matrix inversion), or graphical methods. In Philippine engineering board exams, simultaneous equations appear both as direct mathematical problems and embedded in word problems (mixture, motion, work, age, and investment problems) where multiple conditions must be satisfied simultaneously.

175. Slope

The ratio of the change in y to the change in x for a line: m = (y₂ − y₁)/(x₂ − x₁). The slope measures the steepness and direction of a line. A positive slope indicates an increasing line, negative slope a decreasing line, zero slope a horizontal line, and undefined slope a vertical line. Slope is fundamentally an algebraic concept, though it bridges directly to analytic geometry. It also connects to the tangent function and to the rate of change concept in calculus.

176. Solution Set

The set of all values of the variable that satisfy a given equation or inequality. For an equation, the solution set may contain one value, several values, infinitely many values, or no values at all (the empty set). For an inequality, the solution set is typically an interval or union of intervals, expressed using interval notation or set builder notation. On the board exam, correctly expressing the solution set in the format the question requires, whether as a list of values, an interval, or a graph on the number line, is part of giving a complete answer.

177. Square Matrix

A matrix with the same number of rows and columns (n × n). Only square matrices have determinants, inverses, eigenvalues, and traces. The most common square matrices in board exam problems are 2×2 and 3×3. The main diagonal of a square matrix runs from the upper left to the lower right. Special square matrices include the identity matrix, diagonal matrix, symmetric matrix, skew symmetric matrix, and triangular matrix.

178. Square Root

The value that, when multiplied by itself, gives the original number. Written as √x, it is the principal (non negative) square root of x. For x ≥ 0, √x × √x = x. The square root function has domain [0, ∞) and range [0, ∞). In solving quadratic equations by taking square roots, remember that x² = k gives x = ±√k (both positive and negative roots), not just the positive root. Missing the negative root is a common board exam error.

179. Substitution Method

A method for solving a system of equations by solving one equation for one variable and substituting the resulting expression into the other equation(s). The substitution method is most efficient when one of the equations has a variable with a coefficient of 1 or −1, making isolation straightforward. For systems of nonlinear equations (such as a line and a parabola), substitution is usually the preferred method. The result after substitution is a single variable equation that can be solved directly.

180. Sum of Cubes

The factoring identity a³ + b³ = (a + b)(a² − ab + b²). This is the companion to the difference of cubes formula. The factor (a + b) is the linear factor and (a² − ab + b²) is the quadratic factor. The quadratic factor does not factor further over the real numbers. On the board exam, the sum of cubes appears in factoring problems and in problems involving polynomial roots. Note the sign pattern: the binomial factor has the same sign as the original expression, and the middle term of the quadratic factor is negative.

181. Sum of Roots

By Vieta’s formulas, the sum of all roots of the polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0 equals −aₙ₋₁/aₙ. For a quadratic ax² + bx + c = 0, the sum of the roots is −b/a. The sum of roots formula is one of the most frequently applied Vieta’s formulas on Philippine engineering board exams. Problems that ask for the sum of roots without requiring you to find the individual roots are solved instantly using this formula.

182. Supplementary Equations

This term, as used in some Philippine engineering review materials, refers to the additional equations needed to fully determine a system. In a system with n unknowns, you need at least n independent equations for a unique solution. Supplementary equations provide the additional constraints. This concept is more commonly referred to as “the condition for a unique solution” in formal algebra, but understanding the term helps when reviewing older Filipino engineering board exam reference materials.

183. Surds

Irrational numbers expressed as roots that cannot be simplified to rational numbers. Examples include √2, √3, and ∛5. Operations on surds include simplifying, rationalizing denominators, and adding or subtracting like surds. Rationalizing a denominator means eliminating any radical from the denominator by multiplying numerator and denominator by the conjugate surd (for binomial denominators) or by the appropriate radical (for monomial denominators). Rationalizing is a standard algebraic simplification step on the board exam.

184. Symmetric Matrix

A square matrix A that equals its own transpose: A = Aᵀ, meaning aᵢⱼ = aⱼᵢ for all i and j. Symmetric matrices have real eigenvalues and appear in engineering problems involving energy methods, covariance matrices in statistics, and stiffness matrices in structural analysis. The transpose of a matrix is found by swapping rows and columns. For a symmetric matrix, the entries above and below the main diagonal are mirror images of each other.

185. Synthetic Division

A shorthand method for dividing a polynomial by a linear factor (x − r). It uses only the coefficients of the polynomial and the value r, replacing the full long division algorithm. The process produces the quotient coefficients and the remainder. If the remainder is zero, then r is a root of the polynomial and (x − r) is a factor. Synthetic division is significantly faster than long division for linear divisors and is the standard method used in Philippine engineering board exam solutions involving polynomial division.

186. System of Equations

A set of two or more equations involving the same variables, all of which must be satisfied simultaneously. Systems can be linear (all equations are degree 1) or nonlinear (at least one equation has degree greater than 1). A system of n linear equations in n unknowns has a unique solution (independent), no solution (inconsistent), or infinitely many solutions (dependent), depending on the relationship between the equations. Systems of equations are the algebraic foundation for virtually every engineering analysis problem.

187. Term

A single element in a polynomial or algebraic expression, consisting of a coefficient and one or more variable factors. Terms are separated by addition and subtraction signs. In the polynomial 5x³ − 2x² + 7x − 1, there are four terms. Like terms have the same variables raised to the same powers and can be combined by adding their coefficients. Identifying and combining like terms is a fundamental step in simplifying any algebraic expression.

188. Transpose of a Matrix

The matrix obtained from A by swapping its rows and columns. If A is an m × n matrix, then its transpose Aᵀ is an n × m matrix where element (Aᵀ)ᵢⱼ = aⱼᵢ. Properties of the transpose include: (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, and (AB)ᵀ = BᵀAᵀ (note the reversed order). The transpose is used in defining symmetric matrices, in the formula for the inverse of a matrix, and in solving normal equations in least squares regression.

189. Trinomial

A polynomial consisting of exactly three terms. The most important trinomial in algebra is the general quadratic ax² + bx + c. Factoring quadratic trinomials is one of the most tested skills in Philippine engineering board exams. The standard method is to find two numbers that multiply to ac and add to b, then use these to split the middle term and factor by grouping. Perfect square trinomials and the difference of squares are special cases that factor immediately by recognition.

190. Upper Bound (Roots)

A value U such that no real root of a polynomial is greater than U. The upper bound theorem states: if synthetic division of f(x) by (x − c) with c > 0 produces all non negative numbers in the quotient row (including the remainder), then c is an upper bound for all real roots. The upper bound theorem, combined with the lower bound theorem and the rational root theorem, is used to efficiently limit the search for real roots of polynomials in theory of equations problems.

191. Variable

A symbol, typically a letter, representing an unknown or changing quantity. Variables are the fundamental elements of algebraic expressions and equations. In a function y = f(x), x is the independent variable and y is the dependent variable. In word problems, translating verbal descriptions into algebraic expressions requires assigning variables to the unknown quantities. Choosing an efficient variable assignment (such as letting x represent the simplest unknown) can significantly reduce the complexity of the resulting equations.

192. Variation

A mathematical relationship between variables expressed as an equation involving a constant of variation k. The four types are direct variation (y = kx), inverse variation (y = k/x), joint variation (z = kxy), and combined variation (y = kxⁿ/zᵐ). Variation problems are a staple of Philippine engineering board exams and appear in the form of two step problems: first determine k from given values, then use k to find the unknown value under new conditions. Setting up the correct variation equation is the critical first step.

193. Venn Diagram

A diagram using overlapping circles to represent sets and their relationships (union, intersection, complement). Venn diagrams are used in probability problems to visualize events and their overlaps, and in set theory problems to count elements satisfying various conditions. In Philippine engineering board exams, Venn diagram problems appear in the mathematics and engineering sciences sections and often ask how many elements belong to exactly one set, both sets, or neither set given total counts and overlap counts.

194. Vertex Form

The form of a quadratic function written as f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola. The vertex form is obtained by completing the square on the standard form. It immediately reveals the vertex, the axis of symmetry (x = h), and whether the parabola opens up (a > 0) or down (a < 0). Board exam problems asking for the maximum or minimum value of a quadratic function are solved most directly using the vertex form or the formula h = −b/(2a) and k = f(h).

195. Vieta’s Formulas

A set of equations relating the coefficients of a polynomial to symmetric functions of its roots. For a quadratic ax² + bx + c = 0 with roots r₁ and r₂: r₁ + r₂ = −b/a (sum of roots) and r₁ × r₂ = c/a (product of roots). For a cubic ax³ + bx² + cx + d = 0 with roots r₁, r₂, r₃: r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = −d/a. Vieta’s formulas are among the most heavily tested tools in the theory of equations section of Philippine engineering board exams.

196. Work Problem

A classic category of algebra word problem involving the rate at which people or machines complete a job. The fundamental relationship is: work = rate × time, with total work usually set equal to 1 (one complete job). If person A finishes in a hours and person B finishes in b hours, their combined rate is 1/a + 1/b, and the time to finish together is 1/(1/a + 1/b) = ab/(a + b). Work problems appear in every Philippine engineering board exam and require careful rate setup before solving.

197. Zero Exponent Rule

The rule stating that any nonzero number raised to the power of zero equals 1: a⁰ = 1 for a ≠ 0. This rule is derived from the quotient rule: aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰, and since any nonzero number divided by itself equals 1, a⁰ = 1. Note that 0⁰ is an indeterminate form and is not covered by this rule. The zero exponent rule appears constantly in simplifying exponential expressions and evaluating polynomial functions at specific values.

198. Zero Matrix

A matrix in which every element is zero, denoted O or 0. The zero matrix is the additive identity for matrix addition: A + O = A for any matrix A of the same dimensions. The zero matrix is analogous to the number 0 in ordinary arithmetic. Unlike ordinary arithmetic, however, the product of two nonzero matrices can be the zero matrix (AB = O does not imply A = O or B = O), which is an important distinction in matrix algebra.

199. Zero of a Function

A value of x for which f(x) = 0. Zeros of a function are the x intercepts of its graph. Finding the zeros of a function is equivalent to solving the equation f(x) = 0. For polynomial functions, zeros are the same as the roots of the polynomial equation. The number of real zeros of a polynomial is determined by the degree, the discriminant (for quadratics), Descartes’ rule of signs (for higher degrees), and the intermediate value theorem. Zeros are a central concept throughout the theory of equations.

200. Zero Polynomial

The polynomial in which all coefficients are zero, resulting in f(x) = 0 for all values of x. The zero polynomial has no defined degree (or is sometimes assigned degree −∞ by convention). It is the additive identity for polynomial addition, just as the number 0 is the additive identity for real number addition. In the context of polynomial equations and algebra, the zero polynomial must be distinguished from the zero root (a root equal to zero): the zero polynomial is identically zero everywhere, while a polynomial with a zero root is zero only at x = 0.

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201. Zero Product Property

The property stating that if the product of two or more factors equals zero, then at least one of the factors must equal zero: if ab = 0, then a = 0 or b = 0 (or both). This property is the algebraic basis for solving polynomial equations by factoring. The procedure is: factor the polynomial completely, set each factor equal to zero, and solve. Without the zero product property, factoring would not directly yield the solutions. It is one of the most used properties in all of algebra.

CONCLUSION:

Algebra is the broadest single subject in the Philippine engineering board exam, and breadth is both its challenge and its opportunity. The challenge is that you cannot afford to leave any major area completely unprepared. The opportunity is that algebraic skills are transferable: mastering the theory of equations helps you in trigonometry, mastering logarithms helps you in engineering sciences, and mastering matrices helps you in advanced engineering subjects. Time invested in algebra during your review period pays dividends across every other subject area. The 201 terms in this glossary are the vocabulary you need to read, understand, and solve algebraic problems at the board exam level. If you can define every term here and work a problem for each one, you are in excellent shape.

For your focused review, prioritize the following areas because they appear most consistently across different engineering board exams. First, quadratic equations: the quadratic formula, the discriminant, Vieta’s formulas for sum and product of roots, and the nature of roots. Second, the binomial theorem: finding specific terms in a binomial expansion, especially the rth term formula and identifying the middle term. Third, progressions and series: the nth term and sum formulas for both arithmetic and geometric sequences, and the infinite geometric series formula. Fourth, the theory of equations: the rational root theorem, synthetic division, the remainder theorem, the factor theorem, and Descartes’ rule of signs for polynomials of degree three and four. These areas alone account for a substantial portion of algebra items in every engineering licensure exam.

Beyond those priorities, make sure you are comfortable with logarithm and exponential equations, variation problems (direct, inverse, joint, and combined), systems of linear equations (substitution, elimination, Cramer’s rule, and matrix methods), and the classic word problem types: work, mixture, motion, age, digit, clock, and investment problems. These word problem types appear in every exam, and the examinees who solve them fastest are the ones who have internalized the setup process through repeated practice. Do not just read through this glossary and consider yourself prepared. Use it as a checklist, identify the topics where your practice is weakest, and go solve problems in those areas. That is how you turn vocabulary into exam-ready skill.

If you want to practice applying these concepts, head over to our Algebra Problems and Solutions section here on PinoyBix. We have hundreds of solved problems organized by topic and all with complete solutions so you can see exactly how each concept is used in actual exam-type questions.

Good luck on your review. You’ve got this.