Factoring Trinomials with Leading Coefficient Not 1

If the leading coefficient of a quadratic trinomial is not 1, we can again use grouping to factor the trinomial. However, the procedure is slightly different.

Consider the trinomial 2x² + 11x + 12, for which a = 2, b = 11, and c = 12. First find ac, the product of the leading coefficient and the constant term. In this case ac = 2 · 12 = 24. Now find two integers with a product of 24 and a sum of 11. The pairs of integers with a product of 24 are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Only 3 and 8 have a product of 24 and a sum of 11. Now replace 11x by 3x  8x and factor by grouping:

This strategy for factoring a quadratic trinomial, known as the ac method, is summarized in the following box. The ac method works also when a = 1.

 

Strategy for Factoring ax² + bx + c by the ac-Method

To factor the trinomial ax² + bx + c

1. find two integers that have a product equal to ac and a sum equal to b,

2. replace bx by two terms using the two new integers as coefficients,

3. then factor the resulting four-term polynomial by grouping.

 

Example 1

Factoring ax² + bx + c with a ≠ 1

Factor each trinomial.

a) 2x² + 9x + 4

b) 2x² + 5x - 12

Solution

a) Because 2 · 4 = 8, we need two numbers with a product of 8 and a sum of 9. The numbers are 1 and 8. Replace 9x by x + 8x and factor by grouping:

Note that if you start with 2x² + 8x + x + 4, and factor by grouping, you get the same result.

b) Because 2(-12) = -24, we need two numbers with a product of -24 and a sum of 5. The pairs of numbers with a product of 24 are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. To get a product of -24, one of the numbers must be negative and the other positive. To get a sum of positive 5, we need -3 and 8: