Chapter 8.5, Problem 42P

The area of a rectangular door is given by the trinomial $x^2-14x+45.$

The door’s width is $(x-9).$

What is the door’s length?

To determine

To find : the door’s length.

Answer to Problem 42P

  $x-5$

Explanation of Solution

Given information: The area of a rectangular door is given by the trinomial $x^2-14x+45$. The door’s width is $\left(x-9\right)$ .

Calculation:

We will factorize the polynomial for the area of the rectangular painting by splitting the midle term into two terms whose sum is $-14x$ and product is $45x^2$ .

Therefore, by splitting the middle term , we get

  $\begin{array}{l}{x}^2-14x+45\\ =x^2-9x-5x+45\\ =\left(x^2-9x\right)+\left(-5x+45\right)\\ =x\left(x-9\right)-5\left(x-9\right)\\ =\left(x-9\right)\left(x-5\right)\end{array}$

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

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

length $=\frac{x^2-14x+45}{x-9}=\frac{\left(x-9\right)\left(x-5\right)}{x-9}=x-5$

Thus, length of the door is $x-5$ .