What you will be able to do
- Always check for a greatest common factor first.
- For x² + bx + c, find two numbers with product c and sum b.
- Recognise difference-of-squares structure.
- Multiply the factors to verify.
Factoring reverses multiplication
Expanding changes a product into a sum; factoring changes a sum into an equivalent product. The fastest reliability check is to multiply the factors and recover the original polynomial.
Factoring completely means no factor can be factored further within the expected number system and course scope.
Step 1: remove the greatest common factor
Worked example 1
Factor 6x³ − 9x².
- 1.The numerical GCF is 3.
- 2.Both terms contain x².
- 3.Factor out 3x²: 3x²(2x − 3).
Answer: 3x²(2x − 3)
Check: Distribute 3x² to recover both terms.
Step 2: factor monic trinomials
Worked example 1
Factor x² − x − 12.
- 1.Find two numbers with product −12 and sum −1.
- 2.The numbers are −4 and 3.
- 3.Write (x − 4)(x + 3).
Answer: (x − 4)(x + 3)
Check: Outer and inner terms combine to −x.
Step 3: recognise a difference of squares
Worked example 1
Factor 25x² − 16.
- 1.Write as (5x)² − 4².
- 2.Use a² − b² = (a − b)(a + b).
Answer: (5x − 4)(5x + 4)
Check: Middle terms cancel when expanded.
Use a factoring decision path
Practise quadratic equationsGraph quadratic functions
- 1Arrange terms in descending powers.
- 2Factor out the GCF.
- 3Count terms and identify a familiar structure.
- 4For x² + bx + c, test factor pairs of c.
- 5For a² − b², use conjugate factors.
- 6Expand to verify and factor again if possible.
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Useful answers
Frequently asked questions
How do I know whether factoring is correct?
Expand the factors. The result must exactly match the original polynomial.
Why factor the GCF first?
It simplifies the remaining expression and prevents incomplete final answers.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.