Factoring Polynomials - GCF

Factoring polynomials is the process of breaking down a polynomial into simpler polynomials (called factors) that, when multiplied together, give the original polynomial. Factoring is a key skill in algebra and is used to simplify expressions, solve equations, and analyze functions.

The following diagram shows how to factor polynomials using the Greatest Common Factor (GCF). Scroll down the page for more examples and solutions of factoring polynomials using the GCF.

 

Different Methods to Factor Trinomials & Polynomials

• Factor Trinomials

  Factor Trinomials
  Factor Trinomials by GCF
  Factor Perfect Square Trinomials
  Factor Trinomials by Grouping
  Factor Trinomials by Trail & Error - Unfoiling
  Factor Trinomials with Two Variables

Factor Polynomials Worksheets
Practice your skills with the following Algebra Worksheets.
Printable & Online Algebra Worksheets

Here are some common factoring techniques for polynomials:

1. Greatest Common Factor (GCF):
   Identify the GCF: Find the largest factor that divides evenly into all terms.
   Factor Out the GCF: Divide each term by the GCF and write the expression in factored form.
   Example: 4x³ + 8x² - 12x = 4x(x² + 2x - 3)
   
2. Factoring by Grouping:
   Used for polynomials with four or more terms.
   Group Terms: Group the terms into pairs.
   Factor Each Group: Factor out the GCF from each group.
   Factor the Common Binomial: If both groups share a common binomial factor, factor it out.
   Example: x³ + 3x² + 2x + 6 = (x³ + 3x²) + (2x + 6) = x²(x + 3) + 2(x + 3) = (x + 3)(x² + 2)
   
3. Factoring Trinomials (ax² + bx + c):
   Simple Trinomials (a = 1):
   Find two numbers that multiply to ‘c’ and add to ‘b’.
   Write the factored form as (x + number 1)(x + number 2).
   Example: x² + 5x + 6 = (x + 2)(x + 3)
   Trinomials (a ≠ 1):
   “ac” Method:
   Multiply ‘a’ and ‘c’.
   Find two numbers that multiply to ‘ac’ and add to ‘b’.
   Rewrite the middle term (‘bx’) using these two numbers.
   Factor by grouping.
   Example: 2x² + 7x + 3
   ac = 2 × 3 = 6
   Numbers: 1 and 6
   2x² + 1x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)
   Trial and Error:
   Try different combinations of factors of ‘a’ and ‘c’ until you find the correct combination.
   
4. Difference of Squares (a² - b²):
   Pattern: a² - b² = (a + b)(a - b)
   Example: x² - 9 = (x + 3)(x - 3)
   
5. Sum and Difference of Cubes (a³ ± b³):
   Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
   Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
   Example: x³ + 8 = (x + 2)(x² - 2x + 4)
   
6. Perfect Square Trinomials (a² ± 2ab + b²):
   Pattern: a² + 2ab + b² = (a + b)²
   Pattern: a² - 2ab + b² = (a - b)²
   Example: x² + 6x + 9 = (x + 3)²
   

Factoring Polynomials - GCF (part 1) Factoring Polynomials
A little background information and an introduction to factoring out the greatest common factor.
Factor:
5x + 10
6x + 12

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Factoring Polynomials - GCF (part 2) Factoring Polynomials
More examples for factoring out the greatest common factor.
Factor:
60x³ - 50 x²
84x³y⁴ - 28x⁵y² + 14x²y²

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Factoring Polynomials - GCF (part 3)
More examples and information for factoring out the greatest common factor.
Factor:
-8x - 12
6x - 15
32x³ - 20x
36x⁴y² - 24xy³ + 44x²y²

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Factoring Using the Greatest Common Factor, GCF
Factor out the GCF:
a) 2x³y⁸ + 6x⁴y² + 10x⁵y¹⁰
b) 6a¹⁰b⁸ + 3a⁷b⁴ - 24a⁵b⁶

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