Factoring Polynomials

Core Concept

Factoring is the reverse of distribution. Given a polynomial, you rewrite it as a product of simpler expressions. The factored form is equivalent to the original — they produce the same output for every value of the variable — so you can substitute one for the other anywhere on the test.

Step 0 — always check for a GCF first. Even when a difference-of-squares or trinomial pattern is present, pulling out the GCF first keeps the numbers small.

Strategy 1 — Greatest Common Factor

Find the largest coefficient that divides every term AND the lowest power of each variable that appears in every term. Factor it out using the distributive property.

The GCF here is $3x$: the coefficient $3$ divides $6, 15, 9$, and every term has at least one factor of $x$.

Strategy 2 — Difference of Squares

A binomial of the form $A^2 - B^2$ factors as $(A - B)(A + B)$. Both terms must be perfect squares and the operation must be subtraction (a sum of squares does not factor over the reals).

Strategy 3 — Trinomial Factoring

For $ax^2 + bx + c$ with $a = 1$: find two numbers that multiply to $c$ and add to $b$.

For $ax^2 + bx + c$ with $a \neq 1$: use the AC method — find two numbers that multiply to $ac$ and add to $b$, split the middle term, then factor by grouping.

Using the factored form to answer a question

Many SAT questions don't stop at "factor this." They ask: Which value makes the expression equal zero? What is the constant $k$? What is the simplified form of this fraction? Always ask yourself what the factored form tells you before writing down an answer.