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# 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