the volume of a rectangular prism is $(x^{4}+4x^{3}+3x^{2}+8x + 4)$, and the area of its base is…

the volume of a rectangular prism is $(x^{4}+4x^{3}+3x^{2}+8x + 4)$, and the area of its base is $(x^{3}+3x^{2}+8)$. if the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?\n$x + 1-\frac{4}{x^{4}+4x^{3}+3x^{2}+8x + 4}$\n$x + 1+\frac{4}{x^{4}+4x^{3}+3x^{2}+8x + 4}$\n$x + 1-\frac{4}{x^{3}+3x^{2}+8}$\n$x + 1+\frac{4}{x^{3}+3x^{2}+8}$

the volume of a rectangular prism is $(x^{4}+4x^{3}+3x^{2}+8x + 4)$, and the area of its base is $(x^{3}+3x^{2}+8)$. if the volume of a rectangular prism is the product of its base area and height, what is the height of the prism?\n$x + 1-\frac{4}{x^{4}+4x^{3}+3x^{2}+8x + 4}$\n$x + 1+\frac{4}{x^{4}+4x^{3}+3x^{2}+8x + 4}$\n$x + 1-\frac{4}{x^{3}+3x^{2}+8}$\n$x + 1+\frac{4}{x^{3}+3x^{2}+8}$

Answer

Explanation:

Step1: Recall volume - height - base - area formula

The volume $V$ of a rectangular prism is given by $V = A\times h$, where $A$ is the base - area and $h$ is the height. So, $h=\frac{V}{A}$. Here, $V=x^{4}+4x^{3}+3x^{2}+8x + 4$ and $A=x^{3}+3x^{2}+8$.

Step2: Perform polynomial long - division

Divide $x^{4}+4x^{3}+3x^{2}+8x + 4$ by $x^{3}+3x^{2}+8$: [ \begin{align*} \frac{x^{4}+4x^{3}+3x^{2}+8x + 4}{x^{3}+3x^{2}+8}&=\frac{x^{4}+3x^{3}+x^{3}+3x^{2}+8x + 4}{x^{3}+3x^{2}+8}\ &=x+\frac{x^{3}+3x^{2}+8x + 4}{x^{3}+3x^{2}+8}\ &=x + 1+\frac{4}{x^{3}+3x^{2}+8} \end{align*} ]

Answer:

$x + 1+\frac{4}{x^{3}+3x^{2}+8}$