Chapter 8.6, Problem 17P

The area of a rectangular granite countertop is $12x^2+10x-12.$

The width of the countertop is $2x+3.$

What is the length of the countertop?

To determine

To find : the length of the countertop.

Answer to Problem 17P

  $6x-4$

Explanation of Solution

Given information: The area of a rectangular countertop is $12x^2+10x-12$ . The width of the countertop is $2x+3$ .

Calculation:

We will factorize the polynomial for the area of the rectangular countertoop by splitting the midle term into two terms whose sum is 10x and product is $-144x^2$ .

Therefore, by splitting the middle term , we get

  $\begin{array}{l}12x^2+10x-12\\ =12x^2+18x-8x-12\\ =\left(12x^2+18x\right)+\left(-8x-12\right)\\ =6x\left(2x+3\right)-4\left(2x+3\right)\\ =\left(6x-4\right)\left(2x+3\right)\end{array}$

We know that area of a rectangle is product of its length and width.

Therefore, length of the countertop is the ratio of its area to its width.

length $=\frac{12x^2+10x-12}{2x+3}=\frac{\left(6x-4\right)\left(2x+3\right)}{2x+3}=6x-4$

Thus, length of the countertoop is $6x-4$ .