Question on Factoring

Question

Find all integers b such that the trinomial can be factored into the product of two binomials. (Assume b ≠ 0.

 Enter your answers as a comma-separated list.)

8x² + bx − 18

Answer 1

A polynomial ax²+bx+c can be factored into the product of two binomials if there are two factors of ac whose sum is b.

In this case, given the polynomial 8x²+bx-18, we have a=8 and c=-18, which means that ac=(8)(-18)=-144. Therefore, to factor 8x²+bx-18, we must find two factors of -144 and their sum will give us the value of b.

First, let's find all factor pairs of -144:

1, -144

-1, 144

2, -72

-2, 72

3, -48

-3, 48

4, -36

-4, 36

6, -24

-6, 24

8, -18

-8, 18

9, -16

-9, 16

12, -12

Now, let's use the factor pair of 1 and -144 as an example to factor 8x²+bx-18 into a product of two binomials so that we can find the value of b in this case.

Using the factor pair of 1 and -144, we can rewrite 8x²+bx-18 as 8x²+1x-144x-18. Using the grouping method, this can be rewritten as x(8x+1)-18(8x+1). Since (8x+1) is a common factor for both terms, this can be rewritten as (8x+1)(x-18), which is a product of two binomials that is equal to 8x²-143x-18. Thus, b=-143 in this case, which is the sum of the two factors 1 and -144.

The same concepts apply to the rest of the factors pairs listed: b will equal the sum of the two factors. Therefore, the possible values of b are -143, 143, -70, 70, -45, 45, -32, 32, -18, 18, -10, 10, -7, 7, 0.

Answer 2

When using the ac method, you find all the factors of the product of a (8) and c (-18) whose sum is b. So, you must find all of the factors of 8 and -18 and list their sums as possible answers for c:

-1 x 144, 143

-2 x 72, 70

-3 x 48, 45

-4 x 36,  32

-6 x 24, 18

-8 x 18, 10

-9 x 16,  7

-12 x 12, 0

1 x -144, -143

2 x -72, -70

3 x -48, -45

4 x -36, -32

6 x -24, -18

8 x -18, -10

9 x -16, -7