Question 1122314
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

{{{(-x^3-x^2+12x)/(x+4)}}}


{{{(-1*x(x^2+x-12))/(x+4)}}}


and the division you want to perform as synthetic division is, as if you were checking root of {{{-4}}}.
<pre>
-4  |  1  1  -12
    |    -4   12
    |________________
       1  -3  0
</pre>
This is {{{x-3}}}.


The factorization of the volume expression is then  {{{highlight_green(-1*x(x+4)(x-3))}}}.



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.


-

{{{(d/dx)(-x^3-x^2+12x)}}}

{{{-3x^2-2x+12}}}

{{{-3x^2-2x+12=0}}}

{{{3x^2+2x-12=0}}}


{{{x=(-2+- sqrt(4+144))/6}}}
Must be the positive value.


{{{x=(-2+ sqrt(148))/6}}}


{{{x=(-2+2*sqrt(37))/(2*3)}}}


{{{x=-1/3+sqrt(37)/3}}}-------------------Use this to find the length of x+4; and use x to evaluate the volume there.