Mathematico | Mathematico

A cuboid has a square base of side-length $x + 1$ and a height of

h = 2 x + \frac{2}{x + 1} + 7

Cuboid of height 2x + 2/(x+1) + 7

A fully factorised expression for the volume of the cuboid can be given in the form

(a x + b) (x + c) (x + d), a, b, c, d \in N

Find

(a + b) (c + d)

Hint

Don't expand the $(x + 1)^{2}$ you get for the area of the base. Rather, write it as

(x + 1) (x + 1)

and multiply one of the $(x + 1)$ into the expression for the height. The fraction should cancel leaving you with something manageable.

Solution

The volume is given by

\begin{aligned} (x + 1)^{2} (2 x + \frac{2}{x + 1} + 7) \\ = & (x + 1) [2 x (x + 1) + 2 + 7 (x + 1)] \\ = & (x + 1) (2 x^{2} + 9 x + 9) \\ = & (2 x + 3) (x + 1) (x + 3) \end{aligned}