# Factor by Grouping : A Formula

How To Factor by Grouping using the AC Formula

#### How do you factor when a is not 1?

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.

### Practice Problems

Think of 8x as 2x + 6x
3x² + 2x + 6x + 4

Group the 2 pairs : (3x² + 2x) + (6x + 4)
Remove the common factors:: $$x \color{Red}{(3x + 2)} + 2\color{Red}{(3x + 2)}$$

Rewrite as grouped factors: $$(x + 2) \color{Red}{(3x + 2)}$$

3 & 4

Think of 7x as 3x + 4x
3x² + 3x + 4x + 4

Group the 2 pairs : (3x² + 3x) + (4x + 4)
Remove the common factors:: $$3x \color{Red}{(x + 1)}$$ + 4(x + 1)

Rewrite as grouped factors:$$(3x + 4)\color{Red}{(x + 1)}$$

Use the formula
Product of (a)(c) = (5)(9) = 45
What factors of 45 sum to 18?

3 & 15

Think of 18x as 3x + 15x
5x² + 3x + 15x + 9
Group the 2 pairs: (5x² + 3x) + (15x + 9)
Remove the common factors: $$x \color{Red}{(5x + 3)} + 3 \color{Red}{(5x + 3)}$$
Rewrite as grouped factors: $$(x + 3) \color{Red}{(5x + 3)}$$

Apply our formula
Product of (a)(c) = (2)(3) = 6
What factors of 6 add up to to 5?

3 & 2

Think of 5x as 2x + 3x
2x² + 2x + 3x + 3
Group the 2 pairs: (2x² + 2x) + (3x + 3)
Remove the common factors: 2x(x + 1) + 3(x + 1)
Rewrite as grouped factors: (2x + 3)(x + 1)

Remember our formula
Product of (a)(c) = (5)(6) = 30
What factors of 30 add up to to 13?

3 & 10

Think of 13x as 10x + 3x
5x² + 10x + 3x+ 6
Group the 2 pairs :(5x² + 10x) + (3x+ 6)
Remove the common factors: 5x(x + 2) + 3(x + 2)
Rewrite as grouped factors: (5x + 3)(x + 2)

You know the deal-- Use our formula for factoring by grouping
Product of (a)(c) = (7)(2) = 14
What factors of 14 add up to to 9?