## Precalculus (6th Edition) Blitzer

The volume of the provided figure as a polynomial function in standard form is: $V\left( x \right)={{x}^{3}}+7{{x}^{2}}-3x$.
Consider the provided figure: There are two rectangles -- one is the outer rectangle and the second is the inner rectangle which is removed from the outer rectangle. The length, width and height of the outer rectangle is $l=x,w=x+3$ and $h=2x-1$. The length, width and height of the outer rectangle is $l=x,w=x$ and $h=\left( 2x-1 \right)-\left( x+1 \right)$. Now, the volume of the provided figure is the difference of the volume of both the outer and inner rectangles. \begin{align} & V\left( x \right)=\left[ \left( x \right)\left( 2x-1 \right)\left( x+3 \right) \right]-\left[ \left( x \right)\left( x \right)\left[ \left( 2x-1 \right)-\left( x+1 \right) \right] \right] \\ & =\left( x \right)\left( 2{{x}^{2}}+5x-3 \right)-{{x}^{2}}\left( x-2 \right) \\ & =2{{x}^{3}}+5{{x}^{2}}-3x-{{x}^{3}}+2{{x}^{2}} \\ & ={{x}^{3}}+7{{x}^{2}}-3x \end{align} Hence, the volume of the provided figure as a polynomial function in standard form is $V\left( x \right)={{x}^{3}}+7{{x}^{2}}-3x$.