Answer
The factorization of the polynomial: $4x(x-3)(x-4)$,
is equivalent to the volume.
Work Step by Step
Factoring the given polynomial, $4x^{3}-28x^{2}+48x$,
we first note that the gcf = $4x$. Factoring it ou,
$= 4x(x^{2}-7x+12)$
... now, searching for factors of 12 whose sum is $-7$,
... we find $-3$ and $-4$
$= 4x(x-3)(x-4)$
Given the dimensions of the box (left in the image), we use
Volume = length$\times$width$\times$height
Length = $8-2x = $
... gcf =$-2$ ...
$= -2(-4+x)=-2(x-4)$
Width = $6-2x = $
... gcf =$-2$ ...
$= -2(-3+x)=-2(x-3)$
Height = $x$
Volume = $-2(x-4)\cdot[-2(x-3)]\cdot x$
$=(-2)(-2)x(x-4)(x-3)$
$=4x(x-4)(x-3)\qquad $... multiplication is commutative,
$= 4x(x-3)(x-4)$
which is equivalent to the factorization of the polynomial,
