Greatest common factor first
Find the largest factor shared by every term and divide each term by it.
Example6x³ + 9x² = 3x²(2x + 3).
Chapter 08 of 10
Factoring
How can a polynomial be rewritten as a product?
Factoring reverses multiplication. A correct factorization multiplies back to the original expression. Always look for a greatest common factor first, then identify patterns such as trinomials or a difference of squares.
The big ideas
Understand these before you worry about speed.
Factor x² + bx + c
Find two numbers whose product is c and whose sum is b.
Examplex² + 7x + 12 = (x + 3)(x + 4).
Difference of squares
A² − B² factors as (A − B)(A + B). A sum of squares does not factor this way over the real numbers.
Example9x² − 25 = (3x − 5)(3x + 5).
Worked example
Follow the reason for each step—not just the symbols.
Problem
Factor completely 2x³ − 18x.
- Factor the GCF 2x: 2x(x² − 9).
- Recognize a difference of squares: x² − 3².
- Apply A² − B² = (A − B)(A + B).
Answer2x(x − 3)(x + 3)
Try it yourself
Work on paper first. Open each answer only when you are ready to check.
1
Factor 8x² + 12x.
Check answer
4x(2x + 3)2
Factor x² + 9x + 20.
Check answer
(x + 4)(x + 5)3
Factor x² − 2x − 15.
Check answer
(x − 5)(x + 3)4
Factor 16y² − 81.
Check answer
(4y − 9)(4y + 9)Practice and tools
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