Question

A treasure chest has a volume of x3 + 9x2 + 26x + 24 cm3 and the height is x + 4 cm. Find the polynomial that would represent the area of the bottom of the treasure chest? Explain your reasoning.

Answer 1

Presumably, the treasure chest is a rectangular prism. The volume of such a solid is given by the area of the base \(A\) time the height \(H\), i.e.
\[V=AH\]
You're told that the height is \(x+4\), and that the total volume of the chest is \(x^3+9x^2+26x+24\), giving the equation,
\[x^3+9x^2+26x+24=A\left(x+4\right)\]
where \(A\) is a polynomial in terms of \(x\). To find this polynomial, you solve for \(A\):
\[A=\frac{x^3+9x^2+26x+24}{x+4}\]
From here, you can either use long or synthetic division to find and expression for \(A\).