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.

Question

anonymous · Answer

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 $A$.