SOLUTION: Use the Factor Theorem to find all real zeros for the given polynomial function and one factor. (Enter your answers as a comma-separated list.) f(x) = 8x3 + x2 − 55x + 42; x

Algebra ->  Equations -> SOLUTION: Use the Factor Theorem to find all real zeros for the given polynomial function and one factor. (Enter your answers as a comma-separated list.) f(x) = 8x3 + x2 − 55x + 42; x       Log On


   

Question 1157886: Use the Factor Theorem to find all real zeros for the given polynomial function and one factor. (Enter your answers as a comma-separated list.)
f(x) = 8x3 + x2 − 55x + 42; x + 3
Answer by greenestamps(13208) About Me  (Show Source):
You can put this solution on YOUR website!


Strange wording of the problem -- it asks to FIND one factor; but it already SHOWS one factor....

(1) Show that x+3 is a factor using synthetic division:

  -3  |  8    1  -55   42
      |
      +-------------------

  -3  |  8    1  -55   42
      |     -24
      +-------------------
         8  -23

  -3  |  8    1  -55   42
      |     -24   69
      +-------------------
         8  -23   14

  -3  |  8    1  -55   42
      |     -24   69  -42
      +-------------------
         8  -23   14    0

(x+3) is a root; and

8x%5E3%2Bx%5E2-55x%2B42+=+%28x%2B3%29%288x%5E2-23x%2B14%29

There are many methods for factoring that quadratic; I am old school and think the best thing to do is try to use logical reasoning to find the pair of linear factors.

With this one, that leads quickly to the correct factorization.

We have two choices for the form of the factorization:
(4x-?)(2x-)
(8x-?)(x-?)

But logical reasoning quickly tells me the first is impossible, because the middle term of the product is going to be even.

So the factorization is of the form (8x-?)(x-?); quick trial with the factors of 14 leads us to (8x-7)(x-2).

The complete factorization is

8x%5E3%2Bx%5E2-55x%2B42+=+%28x%2B3%29%288x-7%29%28x-2%29

The roots are -3, 7/8, and 2.