A+ A− B

How do you factor when a is not 1?

Factor by Grouping : A Formula

How To Factor by Grouping using the AC Formula

To factor a polynomial you must reduce the polynomial expression to its factors.

We are going to use a method known as the 'ac' method to factor by grouping. Many people do not realize that there is actually a systematic way to factor by grouping. This is the closes thing to a 'formula' that you will find for factoring by grouping. It is always much easier to look at some example problems before reading generalized steps, but the steps go as follows

Formula for Factoring By Grouping

If you have a quadratic equation in the form $$ \color{Red}{a}x^2 + \color{purple}{b}x + \color{Yellow}{c} $$
  • Step 1) Determine the product of AC (the coefficients in a quadratic equation)
  • Step 2) Determine what factors of $$ \color{Red}{a} \cdot \color{Yellow}{c} $$ sum to $$ \color{purple}{b}$$.
  • Step 3) "ungroup" the middle term to become the sum of the factors found in step 2
  • Step 4) group the pairs.
As I expressed earlier, it's much easier to understand this method by simply walking through a few examples. So don't worry if the steps above seem like algebraic nonsense -- just check out the example problems below.

Example Problems

$$ \color{Red}{3}x^2 + \color{purple}{8}x + \color{Yellow}{4} $$
Product of (a)(c) = (3)(4) = 12
What factors of 12 sum to 8?

$$ \color{Red}{3}x^2 + \color{purple}{7}x + \color{Yellow}{4} $$
Product of (a)(c) = (3)(4) = 12
What factors of 12 sum to 7?

$$ \color{Red}{5}x^2 + \color{purple}{18}x + \color{Yellow}{9} $$
Show Factors By Grouping

2x² + 5x + 3
Factor it
5x² + 13x + 6
Factor By Grouping

7x² + 9x + 2
Factor By Grouping