Question 1157886
<br>
Strange wording of the problem -- it asks to FIND one factor; but it already SHOWS one factor....<br>
(1) Show that x+3 is a factor using synthetic division:<br><pre>

  -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</pre>
(x+3) is a root; and<br>
{{{8x^3+x^2-55x+42 = (x+3)(8x^2-23x+14)}}}<br>
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.<br>
With this one, that leads quickly to the correct factorization.<br>
We have two choices for the form of the factorization:
(4x-?)(2x-)
(8x-?)(x-?)<br>
But logical reasoning quickly tells me the first is impossible, because the middle term of the product is going to be even.<br>
So the factorization is of the form (8x-?)(x-?); quick trial with the factors of 14 leads us to (8x-7)(x-2).<br>
The complete factorization is<br>
{{{8x^3+x^2-55x+42 = (x+3)(8x-7)(x-2)}}}<br>
The roots are -3, 7/8, and 2.<br>