Chapter 8.6, Problem 35PPS

To determine

To find: An expression for the perimeter of a rectangle with the given area.

Answer to Problem 35PPS

  $4\left(x+12\right)$

Explanation of Solution

Given:

  $x^2+24x-81$

Calculation:

The area of rectangle is given by $x^2+24x-81$

Find the factors of this expression

Rewrite the middle term as $24x=27x-3x$ . So, the polynomial becomes

  $x^2+27x-3x-81$

Make group as shown below

  $\left(x^2+27x\right)+\left(-3x-81\right)$

Factor out the GCF from each of the group

  $x\left(x+27\right)-3\left(x+27\right)$

Finally, factored out the common factor

  $\left(x+27\right)\left(x-3\right)$

Now, the area of rectangle is given by

  $A=lw$

Comparing this with the above factored form of area to find the expression for the length and width of the rectangle

  $\begin{array}{l}l=\left(x+27\right)\\ w=\left(x-3\right)\end{array}$

Therefore, the perimeter of the rectangle is given by the expression

  $\begin{array}{l}2\left(l+w\right)\\ =2\left(x+27+x-3\right)\\ =2\left(2x+24\right)\\ =4\left(x+12\right)\end{array}$