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

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

Answer

Explanation:

Step1: Recall volume - base - height relation

We know that $V = B\times h$, where $V$ is volume, $B$ is base - area and $h$ is height. So, $h=\frac{V}{B}$.

Step2: Substitute given polynomials

Given $V=x^{4}+4x^{3}+3x^{2}+8x + 4$ and $B=x^{3}+3x^{2}+8$. Then $h=\frac{x^{4}+4x^{3}+3x^{2}+8x + 4}{x^{3}+3x^{2}+8}$.

Step3: Perform polynomial long - division

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

Answer:

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