Almost every AMC algebra problem eventually reduces to simplifying or factoring an expression. The fastest solvers aren't doing more algebra than everyone else — they're recognizing a handful of standard patterns instantly instead of grinding through expansion by hand. This lesson covers those patterns.
Order of operations
Parentheses, exponents, multiplication/division (left to right), addition/subtraction (left to right). On the AMC this rarely trips people up directly, but it matters when you're rewriting a complicated expression under time pressure — a dropped set of parentheses is a free wrong answer.
Identities to recognize instantly
These five patterns show up constantly. Memorize the shapes, not just the formulas:
A less common but occasionally decisive one is the Sophie Germain identity, useful whenever you see a sum of two fourth powers:
And when nothing fits a named pattern, try grouping: split the expression into pairs of terms that each share a common factor, then factor out what's left in common.
Worked example
Problem: Factor completely over the integers.
Solution: Treat as a single variable, say . The expression becomes , a plain trinomial. We need two numbers that multiply to 36 and add to −13: those are −4 and −9. So . Substituting back, and are both differences of squares, so the full factorization is .
The key move — substituting to turn a higher-degree expression into a familiar quadratic shape — is one of the most reusable tricks in competition algebra. Watch for it whenever you see only even powers of a variable.
Practice problems
1.Factor completely.
2.Factor completely.
3.Factor .
4.Factor completely over the integers.