Algebra
Expressions
Factoring expressions
18 practice questions
2 video lessons
Theory + worked examples
Theory
Factoring rewrites an expression as a product — the reverse of expanding. Work through a toolkit:
- GCF first: pull out the greatest common factor.
- Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
- Trinomial: \(x^2+bx+c=(x+p)(x+q)\) with \(pq=c,\ p+q=b\).
Check by expanding — the product should return the original.
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.
- Two terms? Try difference of squares.
- Three terms? Find factors of \(c\) that add to \(b\).
- Check by expanding.
Example 1 — Greatest common factor
Factor \(6x+9\).
Solution
Pull out the GCF \(3\).
| \(6x+9\) | \(=\) | \(3(2x+3)\) |
Example 2 — Difference of squares
Factor \(x^2-9\).
Solution
Recognize \(a^2-b^2\).
| \(x^2-9\) | \(=\) | \((x-3)(x+3)\) |
Example 3 — Trinomial
Factor \(x^2+5x+6\).
Solution
Find two numbers multiplying to \(6\), adding to \(5\).
| \(x^2+5x+6\) | \(=\) | \((x+2)(x+3)\) |
Example 4 — GCF then check
Factor \(2x^2+6x\).
Solution
Take out the common \(2x\).
| \(2x^2+6x\) | \(=\) | \(2x(x+3)\) |
Common pitfalls
Take the GCF first.
\(a^2+b^2\) does not factor over the reals.
Check the middle term when factoring a trinomial.
Frequently asked questions
What is factoring?
Rewriting an expression as a product — the reverse of expanding.
What should you factor out first?
The greatest common factor.
Factor \(x^2-16\).
\((x-4)(x+4)\).
Factor \(x^2+7x+12\).
\((x+3)(x+4)\).
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