The polynomial x 3 + 5x 2 - ­57x -­189 expresses the volume, in cubic inches, of a shipping box, and the width is (x+3) in. if the width of the box is 15 in., what are the other two dimensions? ( hint: the height is greater than the depth.)
a. height: 19 in. depth: 5 in
b. height: 21 in. depth: 5 in.
c. height: 19 in. depth: 7 in.
d. height: 21 in. depth: 7 in.

First find expressions for the other two dimensions by dividing the expression for the volume by the expression for the width via either long or synthetic division. We will use synthetic division here.  Hence

-3| 1  5  -57  -189
   |    -3    -6     189
   
     1  2   -63    0

So we have a remainder of 0 which confirms that (x+3) is a factor of
(x³ + 5x² -57x - 189). The quotient we obtained is

        q(x) = x² + 2x -63 = (x + 9)(x - 7)

Thus (x + 9) and (x - 7) are possible expressions for height and depth. However, we were told that the height is greater than the depth, thus
(x + 9) represents the height and (x - 7) represents the depth.

We were given that the width (x + 3) is 15 in.  Thus x = 12.

∴ height is 12 + 9 or 21 in. and the depth is 12 - 7 or 5 in.

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