SOLUTION: 9. The polynomial -x^3 -x^2 + 12x represents the volume of a rectangular aquatic tank in cubic feet. The length of the tank is (x + 4). a. Use synthetic division to help you

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Question 1122314: 9. The polynomial -x^3 -x^2 + 12x represents the volume of a rectangular aquatic tank in cubic feet. The length of the tank is (x + 4).
a. Use synthetic division to help you factor the volume of the polynomial. How many linear factors should you look for?

b. What are the dimensions of the tank?

c. Find the value of x that will maximize the volume of the box.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Volumes are not negative.

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The polynomial -x^3 -x^2 + 12x represents the volume of a rectangular aquatic tank in cubic feet. ...
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If what you wrote is the way it was given to you, then
%28-x%5E3-x%5E2%2B12x%29%2F%28x%2B4%29

%28-1%2Ax%28x%5E2%2Bx-12%29%29%2F%28x%2B4%29

and the division you want to perform as synthetic division is, as if you were checking root of -4.
-4  |  1  1  -12
    |    -4   12
    |________________
       1  -3  0

This is x-3.

The factorization of the volume expression is then highlight_green%28-1%2Ax%28x%2B4%29%28x-3%29%29.


Note, there are three real roots for -x^3-x^2+12x, and you should expect positive values for the volume to be somewhere on the positive x-axis.

-
%28d%2Fdx%29%28-x%5E3-x%5E2%2B12x%29
-3x%5E2-2x%2B12
-3x%5E2-2x%2B12=0
3x%5E2%2B2x-12=0

x=%28-2%2B-+sqrt%284%2B144%29%29%2F6
Must be the positive value.

x=%28-2%2B+sqrt%28148%29%29%2F6

x=%28-2%2B2%2Asqrt%2837%29%29%2F%282%2A3%29

x=-1%2F3%2Bsqrt%2837%29%2F3-------------------Use this to find the length of x+4; and use x to evaluate the volume there.