SOLUTION: https://gyazo.com/00c1e6f263aed37d9b749dea12435602

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Question 1177174: https://gyazo.com/00c1e6f263aed37d9b749dea12435602
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


p%28x%29+=+-6x%5E3-7x%5E2%2Bqx-12 is divisible by 3x-4. Find q.

Two obvious paths to answering the question:
(1) long division, or synthetic division; remainder must be zero
(2) divisible by 3x-4 means x=4/3 is a root; evaluate p(4/3) and solve the resulting equation for q

(1) If the polynomial is divisible by 3x-4, then it is divisible by x-(4/3). Use synthetic division with 4/3.

 4/3 |  -6  -7    q     -12
     |      -8  -20  (4/3)(q-20)
     +--------------------------
        -6 -15  q-20     0

The remainder has to be 0:

%284%2F3%29%28q-20%29-12+=+0
%284%2F3%29%28q-20%29+=+12
q-20+=+12%283%2F4%29+=+9
q+=+29

ANSWER: q = 29

(2) Set p(4/3)=0 and solve for q.

-6%284%2F3%29%5E3-7%284%2F3%29%5E2%2Bq%284%2F3%29-12=0
-6%2864%2F27%29-7%2816%2F9%29%2B%284%2F3%29q-12=0
-128%2F9-112%2F9%2B12q%2F9-108%2F9=0
%28-348%2B12q%29%2F9=0
-348%2B12q=0
12q=348
q+=+348%2F12+=29

ANSWER: q = 29