Algebra
Polynomials
Factoring polynomials
16 practice questions
2 video lessons
Theory + worked examples
Theory
Factoring reverses multiplication. Work through a toolkit:
- GCF first, always.
- Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
- Trinomial: \(x^2+bx+c=(x+p)(x+q)\).
- Grouping for four terms.
Check by expanding your factors.
The factoring toolkit.
Worked factorizations.
Key patterns:
\[a^2-b^2=(a-b)(a+b),\qquad x^2+(p+q)x+pq=(x+p)(x+q)\]
Always take the GCF first.
How to factor
- Remove the GCF.
- Count terms and match a pattern.
- Apply difference of squares, trinomial, or grouping.
- Factor fully and check.
Example 1 — GCF
Factor \(6x^2+9x\).
Solution
Pull out \(3x\).
| \(6x^2+9x\) | \(=\) | \(3x(2x+3)\) |
Example 2 — Trinomial
Factor \(x^2+7x+12\).
Solution
Two numbers multiplying to \(12\), adding to \(7\).
| \(x^2+7x+12\) | \(=\) | \((x+3)(x+4)\) |
Example 3 — Difference of squares
Factor \(x^2-25\).
Solution
Recognize \(a^2-b^2\).
| \(x^2-25\) | \(=\) | \((x-5)(x+5)\) |
Example 4 — GCF then trinomial
Factor \(2x^2+10x+12\).
Solution
Take out \(2\), then factor.
| \(2(x^2+5x+6)\) | \(=\) | \(2(x+2)(x+3)\) |
Common pitfalls
Take the GCF first.
\(a^2+b^2\) does not factor over the reals.
Factor completely — check each factor.
Frequently asked questions
What should you factor out first?
The greatest common factor.
Factor \(x^2-9\).
\((x-3)(x+3)\).
Factor \(x^2+5x+6\).
\((x+2)(x+3)\).
How do you check a factorization?
Expand the factors to get the original.
More in Polynomials