Expression structure

Factoring Polynomials

Factor polynomials step by step using GCF, grouping, trinomials, and special products, then multiply factors back to confirm the expression.

How this page is maintained

Written for learners, checked against the sources below, and reviewed every year. Last reviewed July 22, 2026.

Part of a complete course

Algebra 1 · Chapter 6: Polynomials and Factoring

Short answer

Factor by pulling out the greatest common factor first, then apply patterns such as difference of squares or trinomial factoring. Verify by multiplying factors back to the original polynomial.

  • Always check for a common factor before anything else.
  • Pattern recognition speeds factoring but still requires verification.
  • Refactoring after expansion is the best correctness check.

Apply a repeatable factoring order

A reliable order is: factor out GCF, look for special products, then factor remaining trinomials or use grouping. This prevents missed factors and partial answers.

Difference of squares factors as a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). Perfect square trinomials and sum or difference of cubes also have known patterns.

Factor trinomials with care

For x2+x^2 + bx + c, find two numbers that multiply to c and add to b. For ax2+ax^2 + bx + c with a not equal to 1, methods such as the AC approach or decomposition by grouping are common.

  • Track signs using product and sum logic.
  • Write candidate factor pairs before selecting one.
  • Multiply back to verify every term.

Factor a quadratic with leading coefficient

A geometry expression gives area polynomial 6x2+11x+36x^2 + 11x + 3.

  1. Compute a*c = 6*3 = 18 and find pair summing to 11: 9 and 2.
  2. Split middle term: 6x2+9x+2x+36x^2 + 9x + 2x + 3.
  3. Group: 3x(2x + 3) + 1(2x + 3).
  4. Factor common binomial: (3x + 1)(2x + 3).
Result: The polynomial factors as (3x + 1)(2x + 3).

Common mistakes

  • Skipping GCF and leaving factors incomplete.
  • Using factor pairs that match product but not sum.
  • Losing sign accuracy when splitting middle terms.
  • Not multiplying factors back for confirmation.

Try one

Factor x2−16x^2 - 16.

It is a difference of squares: (x - 4)(x + 4).

Sources

Growth modelsExponential and logarithmicCore algebra skillLinear equationsVisual algebraGraphing lines

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