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]