Skip to main content
Mathovia

Algebra / Preliminaries

Factoring Polynomials

Factoring is multiplication in reverse: rewriting a polynomial as a product of simpler factors. This lesson covers the greatest common factor, the special patterns from the previous lesson run backward, and the two standard methods for factoring a trinomial.

Practice Problems
Written by
MPhil in Mathematics, Founder of Mathovia
Updated

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Chapter 1 · Lesson 13 of 21Preliminaries
  1. 1.Algebra Preliminaries
  2. 2.Integer Exponents
  3. 3.Negative Exponents
  4. 4.Exponent Rules
  5. 5.Rational Exponents
  6. 6.Radicals
  7. 7.Simplifying Radicals
  8. 8.Rationalizing Denominators
  9. 9.Polynomials
  10. 10.Adding Polynomials
  11. 11.Subtracting Polynomials
  12. 12.Multiplying Polynomials
  13. 13.Factoring Polynomials
  14. 14.Greatest Common Factor
  15. 15.Factoring by Grouping
  16. 16.Factoring Quadratics
  17. 17.Factoring Higher-Degree Polynomials
  18. 18.Rational Expressions
  19. 19.Simplifying Rational Expressions
  20. 20.Complex Numbers
  21. 21.Complex Number Arithmetic
See all algebra chapters →

Every pattern in this lesson is the reverse of one from Polynomials. Multiplying gives ; factoring runs that exact process backward, starting from and recovering . The special product patterns you already know for expanding are exactly the patterns worth recognizing for factoring too.

Factoring Polynomials — key formula
Key formula

What Is Factoring?

Factoring means rewriting a polynomial as a product of simpler polynomials, called factors. A polynomial is factored completely when none of its factors can be broken down any further — no common factor left to pull out, and no recognizable pattern left hiding inside a factor.

Factoring Out the GCF

The first move on every factoring problem, no exceptions: find the greatest common factor (GCF) of every term, and factor it out using the distributive property in reverse.

Here the GCF is — the largest expression that divides evenly into both and . Skipping this step doesn’t make later factoring wrong, but it does make the numbers bigger and the patterns harder to spot.

Properties of Factoring

PatternNameStatement
GCF
Difference of Squares
Perfect Square Trinomial(sum)
Perfect Square Trinomial(difference)
Trinomial, leading coefficient 1

The last row is the key to factoring : find two numbers that multiply to and add to — those two numbers are exactly and .

Example 1: Factoring Out the GCF

ExpressionGCFFactored

Example 2: Factoring Special Patterns

ExpressionPatternFactored
Difference of squares
Difference of squares
Perfect square trinomial
Perfect square trinomial

Example 3: Factoring x² + bx + c

Find two numbers that multiply to and add to .

ExpressionTwo NumbersFactored

Worked Example A: GCF, Then a Pattern

Factor completely.

Step 1 — factor out the GCF:

Step 2 — recognize the difference of squares inside:

Final answer: . Stopping after Step 1 would be a common, incomplete answer — always check what’s left inside the parentheses for another pattern.

Worked Example A: GCF first, then a difference-of-squares pattern.
Worked Example A: GCF first, then a difference-of-squares pattern.

Worked Example B: The ac Method

Factor , where the leading coefficient isn’t 1.

Step 1 — multiply and , then find two numbers multiplying to that product and adding to :

Step 2 — split the middle term using those two numbers:

Step 3 — factor by grouping, pulling a common factor from each pair:

Check by expanding: . ✓

Worked Example B: the ac method, laid out in the box.
Worked Example B: the ac method, laid out in the box.

Worked Example C: GCF, Then a Trinomial

Factor completely.

Step 1 — factor out the GCF:

Step 2 — factor the trinomial: find two numbers multiplying to and adding to :

Step 3 — write the factored trinomial:

Final answer: .

Worked Example C: GCF 4x first, then the trinomial factors.
Worked Example C: GCF 4x first, then the trinomial factors.

Common Mistakes to Avoid

  • Skipping the GCF. Missing a common factor doesn’t make the rest wrong, but it often hides the pattern that would have made the problem simple.
  • Sign errors on the two-numbers step. For , the numbers need a product of (one positive, one negative) and a sum of — check both conditions, not just one.
  • Assuming factors like . It doesn’t; a sum of squares has no real-number factorization.
  • Stopping before it’s complete. still has a difference of squares inside — always re-check every factor for one more pattern.
  • Never checking the answer. Multiplying the factors back out catches sign and arithmetic errors in seconds.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Factor .

Show answer

Answer:

Problem 2. Factor .

Show answer

Answer:

Problem 3. Factor .

Show answer

Answer:

Problem 4. Factor .

Show answer

Answer:

Problem 5. Factor .

Show answer

Two numbers multiplying to and adding to : and .

Answer:

Problem 6. Factor .

Show answer

Two numbers multiplying to and adding to : and .

Answer:

Problem 7. Factor using the ac method.

Show answer

Step 1 — multiply and , find two numbers multiplying to that and adding to :

Step 2 — split and group:

Answer:

Problem 8. Factor completely.

Show answer

Step 1 — factor out the GCF:

Step 2 — factor the difference of squares:

Answer:

Problem 9. Factor completely.

Show answer

Step 1 — factor out the GCF:

Step 2 — factor the trinomial (numbers multiplying to , adding to ): and :

Answer:

Problem 10. Factor .

Show answer

Answer:

Quick Reference

SituationMove
Every term shares a common factorFactor out the GCF first, always
or or
Find two numbers multiplying to , adding to
, The ac method: split the middle term, factor by grouping
Does not factor over the real numbers

These patterns become the main tool in Rational Expressions, where factoring is exactly what simplifies, multiplies, and divides fractions built from polynomials. Polynomials is worth a second look if any pattern above didn’t quite click. From here, the Factoring Practice Problems offers unlimited practice, and both the Algebra lessons list and the site’s calculators keep expanding.

Frequently Asked Questions

What does "factor completely" mean?+

It means every factor in your answer is itself unfactorable — no common factor left to pull out, no difference of squares or trinomial pattern left hiding inside. factored as is only half done; is complete.

Why do I always check for a GCF first?+

Because it makes every factor afterward smaller and easier to work with, and skipping it is the single most common reason a "fully factored" answer turns out not to be. factors as ; trying to apply the difference-of-squares pattern to the original expression directly is still possible but much messier.

Why doesn't a squared plus b squared factor the way a squared minus b squared does?+

factors because it can be written as a product that expands back correctly: . No pair of real-number binomials multiplies out to — there's no sign combination that cancels the cross terms and leaves both squares positive. It stays unfactored over the real numbers.

What is the ac method?+

A way to factor when : multiply and , find two numbers that multiply to that product and add to , use them to split the middle term into two terms, then factor by grouping. See Worked Example B for the full walkthrough.

How do I check that my factoring is correct?+

Multiply the factors back out. If you don't get the original polynomial exactly, something went wrong — this takes thirty seconds and catches nearly every sign or arithmetic error before it becomes a habit.

What comes after factoring in this chapter?+

Rational Expressions, which uses factoring to simplify, multiply, divide, and combine fractions built from polynomials instead of plain numbers.

Related lessons

Greatest Common FactorFactoring by GroupingFactoring QuadraticsFactoring Higher-Degree Polynomials