Factoring Polynomials — SAT Math Explained
The process of rewriting a polynomial as a product of simpler polynomials or monomials, reversing the process of expansion.
The Core Idea
Factoring is a crucial problem-solving tool — it reduces complex expressions to simpler components. The strategy depends on the structure of the polynomial: always check for GCF first, then look for recognizable patterns.
Factoring Hierarchy
1. GCF: Always factor out the Greatest Common Factor first
2. Binomials: Check for Difference of Squares (a² - b²), Sum/Difference of Cubes
3. Trinomials: Check if a = 1 (simple) or a ≠ 1 (AC method)
4. Four Terms: Try factoring by grouping
5. Always verify by expanding your factored form
Key Patterns
a² - b² = (a + b)(a - b)
a² + 2ab + b² = (a + b)²
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
Split into two groups of two terms, factor each group, then factor out the common binomial
Prime Polynomials
Some polynomials cannot be factored over integers — they are called 'prime' or 'irreducible'. Example: x² + 4 cannot be factored with real numbers.
Common Errors to Avoid
Stopping after pulling out the GCF when further factoring is possible
Making sign errors in difference of squares (it's (a+b)(a-b), not (a-b)(a-b))
Not factoring completely — always check if any factor can be factored further
Practice: Factoring Polynomials
5 SAT-style questions. Select your answer and get an instant explanation.
Factor out the GCF: 6x³ + 9x² - 12x
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