Question

a rectangle has a height of $n^{3}+4n^{2}+3n$ and a width of $n^{3}+5n^{2}$. express the area of the entire rectangle. your answer should be a polynomial in standard form. area =

Explanation:

Step1: Recall area formula

The area \(A\) of a rectangle is \(A =\text{height}\times\text{width}\). Here, \(\text{height}=n^{3}+4n^{2}+3n\) and \(\text{width}=n^{3}+5n^{2}\), so \(A=(n^{3}+4n^{2}+3n)(n^{3}+5n^{2})\).

Step2: Use distributive property

\[

$$\begin{align*} A&=n^{3}(n^{3}+5n^{2})+4n^{2}(n^{3}+5n^{2})+3n(n^{3}+5n^{2})\\ &=(n^{3}\times n^{3}+n^{3}\times5n^{2})+(4n^{2}\times n^{3}+4n^{2}\times5n^{2})+(3n\times n^{3}+3n\times5n^{2}) \end{align*}$$

\]

Step3: Apply exponent - rule \(a^{m}\times a^{n}=a^{m + n}\)

\[

$$\begin{align*} A&=(n^{3 + 3}+5n^{3+2})+(4n^{2+3}+20n^{2 + 2})+(3n^{1+3}+15n^{1+2})\\ &=(n^{6}+5n^{5})+(4n^{5}+20n^{4})+(3n^{4}+15n^{3}) \end{align*}$$

\]

Step4: Combine like - terms

\[

$$\begin{align*} A&=n^{6}+(5n^{5}+4n^{5})+(20n^{4}+3n^{4})+15n^{3}\\ &=n^{6}+9n^{5}+23n^{4}+15n^{3} \end{align*}$$

\]

Answer:

\(n^{6}+9n^{5}+23n^{4}+15n^{3}\)