The AC Method for Factoring Trinomials

The AC method for factoring trinomials is a relatively sophisticated but efficient technique for factoring trinomials when the leading term coefficient is not equal to one. In fact, the larger the coefficient, the more appropriate this technique becomes. 

 

STEP-BY-STEP PROCEDURE

Step 1

Write the terms of the trinomial in descending order of degree, i.e., in the form ax²+bx+c.

 

Step 2

Find the product of a and c.

 

Step 3

Find two factors p and q such that pq=ac and p + q = b.

 
This can be done by trial and error, or by solving the system of equations:
 
pq = ac     (1)
p + q = b  (2)

 

Step 4

Rewrite ax²+bx+c as ax²+px +qx+c

 

Step 5

Factor by grouping.

 

ax²+bx+c = ax²+px +qx+c

                  = x(ax + p) + r(ax + p)

                  = (x + r)(ax + p)

 

where r is a factor common to both q and c such that qx + c = r(ax + p).

EXAMPLE

Step 1

6x²-14x-12

 

Step 2

ac = (6)(-12) = -72

 

Step 3

pq = -72      (1)

p + q = -14  (2)

 

Solving the system:

 

From (2) we have:

p + q = -14

<=> q = -14 -p

 

From (1) we have:

pq = -72

<=> p(-14 - p) = -72

<=> p² + 14p -72 = 0

<=> (p - 4)(p + 18) = 0

 

p can be either 4 or -18. We can find q from (1) or (2). 

Therefore p and q are 4 and -18 respectively; or -18 and 4 respectively. Choose either. We'll arbitrarily choose the former. 

 

Step 4

6x²-14x-12 = 6x²- px + qx -12 

                    = 6x²- 4x + -18x -12 

 

Step 5

6x²-14x-12 = 6x²- 4x + -18x -12 

                     =2x(3x + 2) - 6(3x + 2)

                     = (2x - 6)(3x + 2).