The ac- Method
Another method for factoring trinomials of the type ax2 + bx + c, a ≠ 1 , involves the product, ac, of the leading coefficient a and the last term c. It is called the ac-method. Because it use factoring by grouping, it is also referred to as the grouping method.
We know how to factor the trinomial x2 + 5x + 6. We look for factors of the constant term, 6m whose sum is the coefficient of the middle term, 5. What happens when the leading coefficient is not 1? To factor a trinomial like 3x2 – 10x – 8, we can use a method similar to what we used for x2 + 5x + 6.
That method is outlined as follows.
Important Points
To factor ax2 + bx + c, a ≠ 1 using the ac-method:
a. Factor out a common factor, if any.
b. Multiply the leading coefficient a and the constant c.
c. Try to factor the product ac so that the sum of the factors is b.
That is, find integers p and q such that pq = ac and p + q = b .
d. Split the middle term. That is, write it as a sum using the factors
found in step (c).
e. Factor by grouping.
f. Check by multiplying.