Question

The area of a rectangle is modeled by the polynomial (8x^3-6x^2-5x+3) square units. The width of the rectangle is modeled by the polynomial (x+3/4) units. Showing all work, write a polynomial that represents the length of the rectangle.

Answer 1

Given:

Area of a rectangle = $8x^3-6x^2-5x+3$

Width of the rectangle = $x+\dfrac{3}{4}$

To find:

The polynomial that represents the length of the rectangle.

Solution:

We know that,

Area of a rectangle = Length × width

It can be written as

$A=l\times w$

$l=\dfrac{A}{w}$

On substituting the values, we get

$l=\dfrac{8x^3-6x^2-5x+3}{x+\dfrac{3}{4}}$

$l=\dfrac{8x^3-6x^2-5x+3}{\dfrac{4x+3}{4}}$

Splitting the middle terms, we get

$l=4\times \dfrac{8x^3-8x^2+2x^2-2x-3x+3}{4x+3}$

$l=4\times \dfrac{8x^2(x-1)+2x(x-1)-3(x-1)}{4x+3}$

$l=4\times \dfrac{(x-1)(8x^2+2x-3)}{4x+3}$

Again splitting the middle term, we get

$l=4\times \dfrac{(x-1)(8x^2+6x-4x-3)}{4x+3}$

$l=4\times \dfrac{(x-1)(2x(4x+3)-(4x+3))}{4x+3}$

$l=4\times \dfrac{(x-1)(4x+3)(2x-1)}{4x+3}$

$l=4(x-1)(2x-1)$

On simplification, we get

$l=4(2x^2-x-2x+1)$

$l=4(2x^2-3x+1)$

$l=8x^2-12x+4$

Therefore, the polynomial that represents the length of the rectangle is $8x^2-12x+4$.