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How to Factor Polynomials

Learn how to factor polynomials by finding common factors, grouping, trinomials, special products, common mistakes, troubleshooting, and FAQs.

EducationEdited by Pramod Tiwari
2 min read

Quick Answer

To factor polynomials, first look for a greatest common factor, then check for grouping, difference of squares, perfect square trinomials, or trinomial patterns, and multiply your factors to check.

What You Need

  • Polynomial expression
  • Factoring rules
  • Pencil and paper
  • Multiplication skills
  • Patience for checking

Safety Precautions

  • Always check for a greatest common factor first.
  • Keep signs organized.
  • Write terms in standard order when possible.
  • Multiply factors back to verify.
  • Do not assume every polynomial factors neatly over integers.

Step-by-Step Instructions

  1. Step 1

    Put in standard form

    Arrange terms from highest degree to lowest degree if helpful.

  2. Step 2

    Find common factor

    Factor out the greatest common factor from all terms.

  3. Step 3

    Count terms

    Use two-term, three-term, or four-term strategies based on the expression.

  4. Step 4

    Check special patterns

    Look for difference of squares or perfect square trinomials.

  5. Step 5

    Try trinomial factoring

    For ax squared plus bx plus c, find factors that multiply and add correctly.

  6. Step 6

    Use grouping

    For four terms, group pairs and factor each group.

  7. Step 7

    Check by multiplying

    Expand your answer to confirm it equals the original polynomial.

Practical Example

Example: x² + 5x + 6 factors as (x + 2)(x + 3) because 2 and 3 multiply to 6 and add to 5.

Common Mistakes

  • Skipping the greatest common factor
  • Sign errors
  • Not checking by multiplying
  • Forgetting special patterns
  • Forcing unfactorable expressions
  • Dropping terms

Troubleshooting

No factors seem to work

Check for a common factor first, then consider whether it is prime over integers.

Signs are confusing

Use multiplication and addition targets for the constant and middle term.

Four terms do not group well

Try a different grouping order.

Answer expands incorrectly

Go back to the step where signs or factors changed.

FAQs

What should I check first when factoring?

Always check for a greatest common factor.

How do I know factoring is correct?

Multiply the factors back together.

Can all polynomials be factored?

Not always over integers; some are prime in the context of the class.

What is factoring by grouping?

It means grouping terms, factoring each group, and then factoring a shared binomial.

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