Algebra Topics

Algebra is taught as a sequence, but it's better understood as a small number of ideas that keep reappearing in new clothing. This page is the map: what each topic covers, why it exists, and where it turns up in competition.

Foundations

Expressions and Simplification

Expanding, collecting like terms, handling fractions with variables in them, and rationalizing. Unglamorous and completely load-bearing — most "hard" contest problems become easy once the expression is in a sane form, and most wrong answers are arithmetic slips in this layer.

Factoring

Turning a sum into a product. Common factors, difference of squares, perfect square trinomials, grouping, sum and difference of cubes, and the quadratic patterns. Factoring is the single highest-leverage skill in contest algebra because a factored expression tells you its roots for free. → Ten Must-Know Techniques

Exponents and Radicals

The exponent laws, negative and fractional exponents, and manipulating surds. Fractional exponents are where the two topics turn out to be one topic.

Equations

Linear Equations and Systems

One unknown, then several. Substitution and elimination for small systems; recognizing when a system is dependent or inconsistent; and — a common contest move — spotting when the sum or difference of the equations is far more useful than solving outright.

Quadratic Equations

Factoring, completing the square, and the quadratic formula. The discriminant b² − 4ac tells you how many real roots exist before you compute any of them. Vieta's formulas relate the roots to the coefficients (r₁ + r₂ = −b/a, r₁r₂ = c/a) and let you answer questions about roots without finding them — one of the most reliably useful facts in competitive algebra.

Higher-Degree and Rational Equations

Polynomial equations past degree two, the rational root theorem, and equations with variables in denominators — where checking for extraneous solutions stops being optional.

Systems Beyond Linear

Symmetric systems, substitution tricks, and systems where the intended path is a clever combination rather than brute-force elimination.

Order and Bounds

Inequalities

Solving linear and quadratic inequalities, sign analysis on a number line, and the rule that trips up nearly everyone: multiplying by a negative reverses the direction. At contest level this extends to AM–GM, Cauchy–Schwarz, and the art of proving a bound rather than finding a value.

Absolute Value

Distance on the number line, case-splitting, and absolute value inside equations, inequalities, and functions. A steady source of "I got half the answers" errors.

Functions

Functions and Transformations

Domain and range, composition, inverses, and what adding or multiplying constants does to a graph. Transformations turn a family of problems into a single problem plus bookkeeping.

Exponential and Logarithmic Functions

Growth and decay, the log laws, change of base, and solving equations where the unknown is in the exponent. Logs are just exponents written the other way around; students who internalize that stop memorizing the laws.

Sequences and Series

Arithmetic and geometric sequences, closed forms, sums (finite and infinite), and recursion. Contest sequence problems usually hinge on finding the pattern's closed form rather than grinding out terms.

Competition Topics

These sit past most school syllabi but appear routinely at olympiad level. → Advanced Algebra Topics

Polynomials and Polynomial Identities

The factor and remainder theorems, symmetric functions of roots, and identities that let you evaluate ugly expressions without expanding them.

Complex Numbers

Arithmetic in the complex plane, conjugates, modulus and argument, roots of unity, and De Moivre's theorem. Roots of unity in particular solve problems that look purely real.

Modular Arithmetic

Congruences, remainders, and divisibility. Formally number theory, but algebra contests lean on it constantly for "find the last digit" and "prove divisibility" problems.

Symmetric Sums and Substitution

Recognizing symmetry in a system and exploiting it, plus substitutions that collapse a hard expression into a familiar one.

Functional Equations

Equations where the unknown is a function rather than a number. Standard techniques: substituting specific values, testing for injectivity or surjectivity, and guessing the family of solutions then proving nothing else works.

Matrices and Determinants

Matrix arithmetic, determinants, and solving systems with them. Less common in pure algebra contests, but standard at higher levels.

Working Through It

Reading a topic list is not studying it. The order above is roughly the order of dependency, so the fastest route through is to find the highest point you're solid on and work upward from there — see the Resources library for articles on each, and the Practice & Puzzles archive when you want to test rather than read.