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 Sequence

Another commonly used name for arithmetic progression. Each term differs from the preceding term by a fixed constant called the common difference. The nth term is aₙ = a₁ + (n−1)d. The arithmetic sequence is the discrete analog of a linear function: both are defined by a constant rate of change. Recognizing an arithmetic sequence from a list of numbers and applying the nth term and sum formulas efficiently are the core skills tested on the board exam.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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/decay problems). Manipulating bases correctly is essential for solving exponential equations.

19. 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.

20. 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.

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 Mathematics 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. 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.

24. 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, Boolean algebra is a required topic in digital systems and logic design.

25. Cauchy’s Bound

A theorem providing an upper bound for the absolute values of all roots of a polynomial. For a polynomial with leading coefficient aₙ, Cauchy’s bound states that all roots lie within a circle of radius 1 + max(|aₖ|/|aₙ|) for k from 0 to n−1. This bound is used to limit the search range when looking for real or complex roots. It appears occasionally in advanced theory of equations problems in Mathematics engineering board exams.

26. 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.

27. 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.

28. 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.

29. 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.

30. 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.

31. 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.

32. 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.

33. 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 Mathematics 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.

34. 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.

35. Common Logarithm

A logarithm with base 10, written as log(x) or log₁₀(x). The common logarithm is the standard logarithm in Mathematics 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.

36. 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.

37. 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.

38. 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.

39. 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.

40. 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.

41. 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.

42. 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.

43. 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.

44. 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.

45. 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.

46. 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.

47. 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.

48. Continued Fraction

An expression of the form a₀ + 1/(a₁ + 1/(a₂ + 1/(a₃ + …))) where a₀, a₁, a₂, … are integers. Continued fractions provide exact representations of irrational numbers and are used in number theory applications. While not a common topic in most Mathematics engineering board exams, continued fractions appear occasionally in advanced mathematics problems and in some ECE board exam problems related to number systems and algorithm analysis.

49. 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.

50. 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 Mathematics engineering board exam problems because it gives a direct formula for each variable without requiring full row reduction.

51. 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.

52. 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.

53. 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.

54. 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.

55. 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.

56. 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 Mathematics engineering board exams.

57. Difference of Cubes

The factoring identity a³ − b³ = (a − b)(a² + ab + b²). This is one of the standard special product formulas tested in Mathematics 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.

58. 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.

59. 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.

60. 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.

61. 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.

62. 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.

63. 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 Mathematics engineering board exams.

64. 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.

65. 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.

66. 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.

67. 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 Mathematics engineering board exams.

68. 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.

69. 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.

70. 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.

71. 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).

72. 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/discharging.

73. 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.

74. Extraneous Root

See Extraneous Solution. This term is used interchangeably with extraneous solution, particularly in Mathematics engineering board exam review books when referring to polynomial and radical equations. An extraneous root is a value that satisfies a derived equation but not the original, arising from operations that are not reversible for all values.

75. 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.

76. 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.

77. 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 Mathematics engineering board exam, the factor theorem is one of the most directly tested theorems in the theory of equations.

78. 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.

79. 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 Mathematics 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.

80. 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.

81. 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.

82. 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.

83. 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.

84. 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.

85. 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.

86. 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 Mathematics engineering board exam.

87. Geometric Sequence

Another commonly used name for geometric progression. Each term is obtained by multiplying the preceding term by the constant common ratio r. The nth term is aₙ = a₁ × rⁿ⁻¹. The geometric sequence is the discrete analog of an exponential function: both are defined by a constant multiplicative rate of change. Recognizing a geometric sequence by computing ratios of consecutive terms and applying the nth term and sum formulas efficiently are the core skills tested on the board exam.

88. 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.

89. 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.

90. 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.

91. 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 Mathematics engineering board exams, harmonic progression problems are typically solved by converting to the corresponding arithmetic progression using reciprocals and then applying arithmetic progression formulas.

92. Heron’s Formula

A formula for the area of a triangle given only the lengths of its three sides: Area = √[s(s−a)(s−b)(s−c)], where s = (a + b + c)/2 is the semi-perimeter. Heron’s formula is the standard tool for finding triangle areas when no angle is given. While primarily a geometry formula, it is algebraic in nature and appears in combined algebra-geometry problems on the engineering board exam. Identifying the semi-perimeter correctly is the key preliminary step.

93. 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.

94. 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.

95. 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.

96. 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.

97. 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.

98. 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.

99. 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.

100. 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.

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.