# AC Method of Factoring

## What is the AC method?

The AC method of factoring is used to factor polynomials of the form Ax² + Bx + C.
A, B, and C represent constants and x is the variable.

## Steps to use the AC Method

- Determine AC by multiplying the A term and C term.
- Find two numbers that add to the B term and multiply to AC. (Call the smaller number M and the larger number N)
- Rewrite the original equation as Ax² + Mx + Nx + C.
- Factor the new equation by grouping.
- Perform final factoring by applying the distributive property.

## Step-by-step Example

Let's factor the following quadratic polynomial using the AC method.

3x² + 4x - 4

### 1. Determine AC.

In 3x² + 4x - 4, A = 3, B = 4, C = -4

[show hint]

AC = 3 * (-4) = -12.

[show solution]

[step 1]

### 2. Find two numbers whose sum is B and product is AC.

Call the smaller number **M** and the larger number **N**.

If you have trouble doing this step in your head, list
all the factors of -12 and sum them until you find a pair that sums to 4.

[show hint]

The factors of -12 are as follows:

1 and -12

-1 and 12

2 and -6

-2 and 6

3 and -4

-3 and 4

[show hint]

Here, you will see that M = -2 and N = 6 satisfy the requirement.
-2 + 6 = 4 = B and (-2)*6 = -12 = AC.

[show solution]

[step 2]

### 3. Rewrite the original equation as Ax² + Mx + Nx + C.

Since M = -2 and N = 6 we get 3x² - 2x + 6x - 4.

Recall from step 2, we chose M and N such
that their sum equals B. Thus Bx = (M + N)x and by the distributive property
(M + N)x = Mx + Nx = Bx. We can then replace Bx in the original equation with
Mx + Nx.

[show explanation]

[step 3]

### 4. Factor the new equation by grouping.

Find the common factor of Ax² + Mx and Nx + C.

In this example Ax² + Mx = 3x² - 2x and Nx + C = 6x - 4.

Find a common factor of 3x²-2x and 6x-4.

[show hint]

(3x-2) is a factor of 3x² - 2x and 6x - 4.

After factoring by grouping we have x(3x-2) + 2(3x-2)

[show solution]

[step 4]

### 5. Perform final factoring.

By the distributive property we know that x(3x-2) + 2(3x-2)
is equivalent to (x + 2)(3x - 2).

Now, you're done!

[show solution]

[step 5]