# How do you write a polynomial that represents the volume of a box that is a rectangular prism has the dimensions length x+6, width x-2, and height x-1?

Feb 4, 2015

The volume of a rectangular prism is simply the product of its three dimensions: in your case, the volume of the prism is, given $x$,

$\left(x + 6\right) \left(x - 2\right) \left(x - 1\right)$.

A polynomial is a sum (with some coefficients) of powers of $x$, so, if we expand the product just written, we have

$\left(\left(x + 6\right) \left(x - 2\right)\right) \left(x - 1\right) =$
$\left({x}^{2} - 2 x + 6 x - 12\right) \left(x - 1\right) =$
$\left({x}^{2} + 4 x - 12\right) \left(x - 1\right) =$
${x}^{3} + 4 {x}^{2} - 12 x - {x}^{2} - 4 x + 12 =$
${x}^{3} + 3 {x}^{2} - 16 x + 12$

Which is a polynomial, and expresses the volume of the prism