Algebra Foundations

Foundational

The symbolic toolkit the rest of the exam is written in: factoring, the relationship between roots and coefficients, inequalities, sequences, and functions. Almost every AMC 10 problem leans on this module somewhere, even when the problem is nominally about geometry or counting.

Why this domain matters on the AMC 10

Roughly a third of AMC 10 problems are explicitly algebraic, and nearly every other problem uses algebra as its working language — setting up an equation from a geometry diagram, simplifying a counting expression, or solving for an unknown that a number-theory argument produces. Weakness here doesn't just cost the "algebra problems"; it slows down and destabilizes everything else. It is also the domain where careless manipulation, not conceptual gaps, causes the most point loss — an accurate factorization or a correctly tracked sign is worth as much as knowing an advanced theorem.

Prerequisite path

Algebra Foundations has no prerequisites in this course — it is the entry point. Everything downstream depends on it.

Algebra Foundations Number Theory Counting & Probability Geometry Mixed Problem Solving Hard AMC 10 Early AIME

Modules

Lessons are ordered; each builds on the last. Work through them in sequence the first time, then revisit individual lessons as needed during review.

Key formulas & techniques

Factoring identities

a² − b² = (a − b)(a + b)

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

The single most useful pattern-recognition skill in AMC algebra: spotting a difference of squares or a sum/difference of cubes hiding inside a larger expression, often after a substitution.

Vieta's formulas (quadratic)

If ax² + bx + c = 0 has roots r, s: r + s = −b⁄a, rs = c⁄a

Lets you compute symmetric expressions in the roots — sums, products, sums of squares or reciprocals — without ever solving for the roots themselves. Covered in depth in the upcoming "Roots, Coefficients & Vieta's Formulas" lesson.

AM-GM inequality (two terms)

For a, b ≥ 0: (a + b)/2 ≥ √(ab), with equality iff a = b

The workhorse inequality for AMC-level optimization: it turns "minimize a sum given a fixed product" (or vice versa) into an immediate bound, and the equality condition often pins down the answer directly.

Common traps

Losing a sign or a root during factoring

Dividing both sides of an equation by an expression that could be zero silently discards a solution. Factor instead of dividing whenever the factor could vanish.

Misapplying Vieta's to the wrong normalization

Vieta's formulas assume the polynomial is written with a specific leading coefficient. Forgetting to divide by a in ax² + bx + c is one of the most common silent errors on this topic.

Treating an inequality's equality case as the answer without checking feasibility

AM-GM and similar bounds are only tight when their equality condition is achievable under the problem's actual constraints. Always verify that the equality case is reachable before reporting it as the extremum.

Curated resources

Unit test

All 12 lessons in this domain are built. A 24-question cumulative assessment — two questions per lesson, presented in mixed order rather than grouped by topic — is the honest way to check whether the material holds up once nothing is telling you which lesson a problem came from.

Take the unit test

Domain mastery checklist

Check these off only once true, not as encouragement — each should be genuinely automatic before moving on to Number Theory, Counting & Probability, or Geometry.