Question

the tables below represent the length and width of rectangles:

n	w(n)	l(n)	area a(n)
2	3	7
3	4	10
n	n + 1	3n + 1

write polynomial functions a(n) to represent the area of a rectangle that has width n + 1 and length 3n + 1.

Explanation:

Step1: Recall area formula

The area $a(n)$ of a rectangle is given by $a(n)=l(n)\times w(n)$, where $l(n)$ is the length and $w(n)$ is the width.

Step2: Substitute given expressions

Given $w(n)=n + 1$ and $l(n)=3n + 1$, we have $a(n)=(n + 1)(3n+1)$.

Step3: Expand the product

Using the FOIL - method: $(n + 1)(3n+1)=n\times3n+n\times1+1\times3n + 1\times1=3n^{2}+n+3n + 1$.

Step4: Combine like - terms

$3n^{2}+(n + 3n)+1=3n^{2}+4n + 1$.

Answer:

$a(n)=3n^{2}+4n + 1$