Reference

Glossary

The words this course uses, each in one or two plain sentences. Lessons link here the first time a word appears. Where math has several names for one idea, the course picks one and lists the others under "Avoid" so you don't get confused.

Numbers and expressions · Equations and inequalities · Lines and systems · Polynomials and factoring · Rational expressions and radicals · Quadratics · Functions, exponentials and logarithms

Numbers and expressions

Integer: A whole number or its opposite: …, −2, −1, 0, 1, 2, …
Signed number: A number written with its sign, + or −, showing which side of 0 it is on.
Avoid: positive/negative number
Opposite: The number the same distance from 0 on the other side: the opposite of 5 is −5.
Avoid: negative of
Absolute value: The distance of a number from 0, always zero or positive. Written |a|.
Avoid: magnitude, size
Fraction: A number written a/b: the numerator a over the denominator b (b is not 0).
Numerator: The top of a fraction.
Denominator: The bottom of a fraction. It can never be 0.
Reciprocal: The fraction flipped upside down: the reciprocal of a/b is b/a. A number times its reciprocal is 1.
Avoid: inverse (for fractions)
LCD (least common denominator): The smallest expression that every denominator divides into.
Avoid: common bottom
Exponent: The small raised number telling how many times the base is multiplied: in 2³ the exponent is 3.
Avoid: power (when meaning the small number)
Base: The number or variable being raised to an exponent: in 2³ the base is 2.
Square root: √a is the nonnegative number whose square is a: √25 = 5.
Order of operations: The agreed order for computing: parentheses, exponents and roots, multiply and divide (left to right), add and subtract (left to right).
Avoid: PEMDAS rules (fine as a memory aid)
Variable: A letter that stands for a number.
Avoid: unknown
Coefficient: The number multiplying a variable: in −4x² the coefficient is −4.
Term: One piece of an expression separated by + or −: 3x² − 5x + 1 has three terms.
Constant: A term with no variable, like 7.
Like terms: Terms with exactly the same variable part, like 3x² and −8x². Only like terms can be combined.
Avoid: similar terms
Expression: Numbers, variables, and operations with no equals sign, like 2x + 5.
Evaluate: Replace each variable with a given number and compute the value.
Avoid: plug in
Distributive property: a(b + c) = ab + ac: multiply the outside number by every term inside.
Avoid: distributing, FOIL (FOIL is only for two binomials)

Equations and inequalities

Equation: Two expressions joined by an equals sign, like 2x + 5 = 11.
Solution: A value of the variable that makes an equation or inequality true.
Avoid: answer, root (root is fine for polynomials)
Solve: Find every solution.
Linear equation: An equation where the variable appears only to the first power, like 3x − 4 = 2x + 1.
Literal equation: An equation with several letters, such as a formula, solved for one chosen letter.
Avoid: formula rearranging
Identity: An equation true for every value of the variable (all real numbers are solutions).
Contradiction: An equation true for no value of the variable (no solution).
Inequality: A statement using <, >, ≤ or ≥. Multiplying or dividing both sides by a negative flips the sign.
Interval notation: A way to write a set of numbers using endpoints: ( ) leaves an endpoint out, [ ] includes it, and ∞ always gets ( ).
Compound inequality: Two inequalities joined by "and" (both true, an overlap) or "or" (at least one true, a union).
Union (∪): All numbers in either set. Used for "or".
Extraneous solution: A value you get while solving that does not work in the original equation. Always check rational and radical equations.
Avoid: fake solution

Lines and systems

Ordered pair: A point written (x, y): x is the horizontal position, y the vertical.
Avoid: coordinates
x-intercept: Where a graph crosses the x-axis. Find it by setting y = 0.
y-intercept: Where a graph crosses the y-axis. Find it by setting x = 0.
Slope: How steep a line is: rise over run, m = (y₂ − y₁)/(x₂ − x₁).
Avoid: gradient, steepness
Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
Point-slope form: y − y₁ = m(x − x₁), the line with slope m through the point (x₁, y₁).
Standard form (of a line): Ax + By = C.
Parallel lines: Lines with the same slope (and different intercepts). They never meet.
Perpendicular lines: Lines that meet at a right angle. Their slopes are negative reciprocals: m and −1/m.
System of equations: Two or more equations solved together. A solution makes every equation true at once.
Substitution method: Solve one equation for a variable, then put that expression into the other equation.
Elimination method: Add or subtract equations (after multiplying if needed) so that one variable cancels.
Avoid: addition method
Inconsistent system: A system with no solution (parallel lines).
Dependent system: A system with infinitely many solutions (the same line twice).

Polynomials and factoring

Monomial: A single term like 5x³.
Binomial: A polynomial with two terms, like x + 3.
Trinomial: A polynomial with three terms, like x² + 5x + 6.
Polynomial: A sum of terms with whole-number exponents on the variables.
Degree: The highest exponent in a polynomial: x³ − 2x + 1 has degree 3.
Scientific notation: A number written as a × 10ⁿ with 1 ≤ |a| < 10.
GCF (greatest common factor): The largest factor shared by every term.
Avoid: common factor
Factor: (verb) Rewrite as a product. (noun) One of the things being multiplied.
Factored form: An expression written as a product, like (x + 2)(x − 3).
Prime polynomial: A polynomial that cannot be factored (over the integers), like x² + 1.
Difference of squares: a² − b² = (a + b)(a − b).
Perfect square trinomial: a² + 2ab + b² = (a + b)², or a² − 2ab + b² = (a − b)².
Sum and difference of cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
Zero product property: If a · b = 0, then a = 0 or b = 0. This is why we set each factor equal to 0.

Rational expressions and radicals

Rational expression: A fraction whose top and bottom are polynomials, like (x + 1)/(x − 3).
Avoid: algebraic fraction
Excluded value: A value that makes a denominator 0, so the variable can't equal it.
Avoid: restriction
Complex fraction: A fraction that has fractions inside its numerator or denominator.
Avoid: stacked fraction
Radical: A root expression like √x or ∛x. The number inside is the radicand; the small number is the index.
Radicand: The expression under the radical sign.
Index: The small number on a radical telling which root: ∛ has index 3. No number means 2.
Simplified radical: A radical with no perfect-square factor left inside (for square roots) and no radical in a denominator.
Rational exponent: A fraction exponent: a^(m/n) = (ⁿ√a)^m.
Avoid: fractional exponent
Rationalize the denominator: Rewrite a fraction so no radical is left in the denominator.
Conjugate: The same binomial with the middle sign switched: a + √b and a − √b.
Imaginary unit i: The number with i² = −1, so √−1 = i.
Complex number: A number a + bi, where a is the real part and b is the imaginary part.

Quadratics

Quadratic equation: An equation that can be written ax² + bx + c = 0 with a ≠ 0.
Standard form (of a quadratic): ax² + bx + c = 0, everything on one side, zero on the other.
Square root property: If x² = k, then x = √k or x = −√k (written x = ±√k).
Completing the square: Adding (b/2)² to x² + bx to make a perfect square trinomial.
Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a), the solutions of ax² + bx + c = 0.
Discriminant: b² − 4ac. Positive: two real solutions. Zero: one. Negative: two complex (no real) solutions.
Parabola: The U-shaped graph of a quadratic function.
Vertex: The turning point of a parabola, at x = −b/(2a). It is the minimum or maximum.
Axis of symmetry: The vertical line through the vertex, x = −b/(2a).
Vertex form: f(x) = a(x − h)² + k, with vertex (h, k).

Functions, exponentials and logarithms

Function: A rule that gives exactly one output for each input.
Function notation: f(x) means "the output of f when the input is x". It is not f times x.
Domain: All allowed inputs of a function.
Range: All outputs a function actually produces.
Vertical line test: A graph is a function if no vertical line crosses it more than once.
Transformation: A shift, stretch, or reflection of a basic graph, like f(x − h) + k.
Composition: (f ∘ g)(x) = f(g(x)): put g's output into f.
Inverse function: f⁻¹ undoes f: if f(a) = b then f⁻¹(b) = a. Its graph is f reflected over y = x.
One-to-one: A function where different inputs always give different outputs (passes the horizontal line test). Only these have inverses.
Exponential function: f(x) = a · bˣ with b > 0, b ≠ 1. The variable is in the exponent.
Growth or decay factor: The base b in a · bˣ: b > 1 means growth, 0 < b < 1 means decay.
Natural base e: The number e ≈ 2.718, used in continuous growth: A = Pe^(rt).
Logarithm: log_b(x) is the exponent you put on b to get x: log_b(x) = y means b^y = x.
Avoid: log
Common log: log x means base 10.
Natural log: ln x means base e.
Change of base: log_b(x) = ln x / ln b (or log x / log b).

Definitions follow OpenStax Intermediate Algebra 2e and College Algebra 2e (CC BY 4.0), reworded in plain language.