EDGE Math 8

Factoring Other Trinomials

Trinomials in the Form $a x^{2} + b x + c$

Here is another method in factoring a trinomial of the form $a x^{2} + b x + c .$

Step 1.	Find the product of a and c.
Step 2.	Find two factors of ac whose sum is equal to b.
Step 3.	Express b as a sum of these factors.
Step 4.	Factor the resulting polynomial by grouping.
Note that in step 2, if no factors of ac will have a sum of b, then the given trinomial is prime.

For example, to factor $2 x^{2} + x - 15,$ the coefficients are a = 2, b = 1 and $c = - 15.$ Thus, you know that $a c = - 30.$

You need to find two factors of –30 (one factor is positive and the other is negative) whose sum is 1 (which means that the positive factor must be greater than the negative factor). The desired factors of –30 are 6 and –5. Using these numbers, the given trinomial can be written as follows:

2 x^{2} + x - 15 = 2 x^{2} + (6 x - 5 x) - 15

If you regroup the terms and use factoring by grouping technique, you will get the following:

\begin{array}{l} 2 x^{2} + x - 15 = 2 x^{2} + (6 x - 5 x) - 15 \\ = (2 x^{2} + 6 x) - (5 x + 15) \\ = 2 x (x + 3) - 5 (x + 3) \\ = (x + 3) (2 x - 5) \end{array}

Thus, $2 x^{2} + x - 15 = (x + 3) (2 x - 5) .$