Factoring Trinomials By Grouping

Factoring a trinomial by grouping is a powerful method, especially useful when the leading coefficient (the ‘a’ in ax² + bx + c) is not 1. It transforms a three-term polynomial into a four-term polynomial, which can then be factored by grouping.

When factoring trinomials by grouping, we first split the middle term into two terms. We then rewrite the pairs of terms and take out the common factor.

The following diagram shows an example of factoring a trinomial by grouping. Scroll down the page for more examples and solutions on how to factor trinomials by grouping.

• Factor Trinomials Worksheets

  Printable
  Factoring Trinomials (a = 1)
  Factoring Trinomials (a > 1)
  Factor Perfect Square Trinomials
  

  Factoring Quadratics (a = 1)
  Factoring Quadratics (a > 1)
  Factor Difference of Squares
  Factor Perfect Square Quadratics
  

  Online
  Factor Binomials by Difference of Squares
  Factor Perfect Square Trinomials
  
  Factor Trinomials or Quadratic Equations
  Factor Different Types of Trinomials 1
  Factor Different Types of Trinomials 2
  
  Solve Trinomials using Quadratic Formula
  Find Discriminants of Quadratic Polynomials

The General Form of a Trinomial:
ax² + bx + c

Steps to Factor a Trinomial by Grouping:

1. Find the product ac.
   
2. Find two numbers that multiply to ac and add to b.
   
3. Rewrite the middle term (bx) using the two numbers found in Step 2.
   Replace bx with nx+mx (where n and m are the two numbers you found).
   
4. Group the first two terms and the last two terms.
   Put parentheses around the first two terms and around the last two terms. Be careful with signs if there’s a negative sign in front of the third term.
   
5. Factor out the Greatest Common Factor (GCF) from each group.
   
6. You should find that the binomial remaining inside the parentheses after factoring out the GCF is the same for both groups. If it’s not, check your work.
   
7. Factor out the common binomial.
   
8. The common binomial becomes one factor. The GCFs you pulled out in Step 5 form the other factor.
   
9. Check your answer by multiplying the two binomials you obtained.

Example:
Factor the following trinomial using the grouping method.
x² + 6x + 8

Solution:
Step 1: Find the product ac:
(1)(8) = 8

Step 2: Find of two factors of 8 that add up to 6:
4 and 2

Step 3: Write 6x as the sum of 2x and 4x:
x² + 2x + 4x + 8

Step 4: Group the two pairs of terms:
(x² + 2x) + (4x + 8)

Step 5: Take out the common factors from each group:
x(x + 2) + 4(x + 2)

Step 6: Since the two quantities in parentheses are now identical. That means we can factor out a common factor of (x + 2):
(x + 4)(x + 2)

Example:
Factor the following trinomial using the grouping method.
5x² - 13 x + 6

Solution:
Step 1: Find the product ac:
(5)(6) = 30

Step 2: Find of two factors of 30 that add up to 13:
3 and 10

Step 3: Write -13x as the sum of -3x and -10x:
5x² - 3x - 10x + 6

Step 4: Group the two pairs of terms:
(5x² - 3x) - (10x + 6)

Step 5: Take out the common factors from each group:
x(5x - 3) - 2(5x - 3)

Step 6: Since the two quantities in parentheses are now identical. That means we can factor out a common factor of (x - 2):
(x - 2)(5x - 3)

How to factor trinomials by grouping?

Example:
Factor 12x² + 34x + 10

• Show Video Lesson

Factor Trinomial By Grouping
Factor: 6x² + 15x - 21

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Factor Trinomial, GCF And Then Grouping Method
Factor: -6x² + 60x - 28

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Factoring Trinomials: The Grouping Method
Factor: 8x² + 35x + 12

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How to factor Trinomials using GCF and the Grouping Method?
Factor: 6x² - 3x - 45

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