Algebra Glossary

Terms are grouped by theme rather than alphabetized, because the definitions make more sense next to their neighbours.

Basic Objects

Variable — a symbol standing for a quantity that is unknown or allowed to change.

Constant — a fixed numerical value.

Coefficient — the number multiplying a variable in a term. In −4x³, the coefficient is −4.

Term — a single number, variable, or product of them, separated from its neighbours by + or −.

Like terms — terms with identical variable parts, so they can be combined. 3x² and −7x² are like terms; 3x² and 3x³ are not.

Expression — a combination of terms with no equality claimed.

Equation — an assertion that two expressions are equal.

Identity — an equation true for every value of the variable, such as (a + b)² = a² + 2ab + b².

Inequality — an assertion that one expression is less than, greater than, or not equal to another.

Solution / root — a value of the variable that makes an equation true.

Extraneous solution — a value produced by a legitimate manipulation (usually squaring, or clearing a denominator) that fails in the original equation. Always check back.

Manipulation

Expand — remove brackets by distributing, turning a product into a sum.

Factor — the reverse: write a sum as a product.

Simplify — rewrite in the smallest equivalent form.

Rationalize — remove a radical from a denominator by multiplying by a suitable form of 1.

Substitute — replace a variable, or a whole sub-expression, with something equal to it. The most under-used technique in school algebra and the most-used one in contests.

Complete the square — rewrite ax² + bx + c as a(x + h)² + k, exposing the vertex and making the equation trivially solvable.

Quadratics and Polynomials

Polynomial — a sum of terms in which every variable has a whole-number exponent.

Degree — the highest exponent in a polynomial. Degree 1 is linear, 2 quadratic, 3 cubic, 4 quartic.

Monic — a polynomial whose leading coefficient is 1.

Quadratic formula — x = (−b ± √(b² − 4ac)) / 2a, the solution of ax² + bx + c = 0.

Discriminant — b² − 4ac. Positive means two distinct real roots, zero means one repeated root, negative means two complex roots. It answers "how many solutions" without solving.

Vieta's formulas — the relationships between a polynomial's roots and its coefficients. For a quadratic: the roots sum to −b/a and multiply to c/a. Lets you answer questions about roots without finding them.

Factor theorem — (x − r) divides a polynomial exactly when r is a root of it.

Remainder theorem — dividing a polynomial P(x) by (x − r) leaves remainder P(r).

Rational root theorem — any rational root of a polynomial with integer coefficients has numerator dividing the constant term and denominator dividing the leading coefficient. Turns "find a root" into a short finite search.

Symmetric function — an expression in several variables unchanged by swapping them, such as a + b + c or ab + bc + ca. Contest problems about roots are usually symmetric-function problems in disguise.

Order, Growth, and Sequences

Absolute value — |x|, the distance of x from zero, always non-negative.

AM–GM inequality — the arithmetic mean of non-negative numbers is at least their geometric mean, with equality exactly when all the numbers are equal. The workhorse of contest inequality problems.

Cauchy–Schwarz inequality — a bound relating sums of products to products of sums; the second workhorse.

Exponential function — a function of the form f(x) = a·bˣ, where the variable is in the exponent.

Logarithm — the inverse of exponentiation: log_b(y) = x means exactly bˣ = y.

Arithmetic sequence — successive terms differ by a constant.

Geometric sequence — successive terms differ by a constant ratio.

Series — the sum of a sequence's terms.

Recursion — defining each term of a sequence from earlier ones, as in the Fibonacci sequence.

Closed form — a formula giving the nth term directly, without computing the ones before it.

Functions

Function — a rule assigning exactly one output to each input.

Domain — the set of permitted inputs. Range — the set of resulting outputs.

Composition — applying one function to another's output, written f(g(x)).

Inverse function — the function undoing another; f⁻¹(f(x)) = x.

Injective (one-to-one) — distinct inputs always give distinct outputs.

Surjective (onto) — every value in the target set is actually achieved.

Functional equation — an equation whose unknown is a function rather than a number.

Complex and Modular

Imaginary unit — i, defined by i² = −1.

Complex number — a number of the form a + bi with a and b real.

Conjugate — a − bi is the conjugate of a + bi. Multiplying a complex number by its conjugate gives a real result, which is why conjugates clear complex denominators.

Modulus — the distance of a complex number from the origin, |a + bi| = √(a² + b²).

Roots of unity — the complex solutions of zⁿ = 1, evenly spaced around the unit circle. They make certain sums collapse to nothing, which contests exploit constantly.

De Moivre's theorem — (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, turning powers of complex numbers into multiplication of angles.

Congruence — a ≡ b (mod n) means a and b leave the same remainder on division by n.

Modular arithmetic — arithmetic performed on remainders. The standard tool for last-digit and divisibility problems.

Competition Vocabulary

Olympiad — a high-level mathematics competition, typically requiring full written proofs rather than answers alone.

Proof — a complete logical argument establishing a claim for every case, not a demonstration on examples.

Counterexample — a single case disproving a general claim.

Without loss of generality (WLOG) — a phrase marking that an assumption (an ordering, say) costs nothing because the remaining cases are symmetric to the one being handled.

Lemma — a smaller result proved on the way to the main one.

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