Chapter 8.7, Problem 41PPE

To determine

Tosummarize:The procedure for factoring a difference of two squares giving two examples.

Explanation of Solution

Given information:

The difference of two squares can be factorize.

Formulaused:

  $\left(a-b\right)\left(a+b\right)=a\cdot a+a\cdot b-b\cdot a-b\cdot b$

Summary of the Formula-Difference of Two Squares

Derivation:

For two numbers a and b the product of their sum and difference is given by $\left(a-b\right)\left(a+b\right)=a\cdot a+a\cdot b-b\cdot a-b\cdot b$

  $\Rightarrow \left(a-b\right)\left(a+b\right)=a^2+a\cdot b-b\cdot a-b^2$

Now, let $b\cdot a=a\cdot b$ , then

  $\left(a-b\right)\left(a+b\right)=a^2+a\cdot b-a\cdot b-b^2$

  $\left(a-b\right)\left(a+b\right)=a^2-b^2$

This formula is known the factoring formula for difference of two squares.

Application:

The application of the formula can be realized as follows:

Example 1: Suppose $a=7$ and $b=3$ , then

Left hand side of the formula

  $a^2-b^2=$

  $7^2-3^2=49-9=40$ ,

Right hand side of the formula

  $\left(a-b\right)\left(a+b\right)=\left(7-3\right)\left(7+3\right)=\left(4\right)\left(10\right)=40$ .

Thus, we have $7^2-3^2=\left(7-3\right)\left(7+3\right)$ .

A number that can be written as the difference of two squares can be factorized using this formula as follow:

Example 2: Factorization of number 112?

  $112=121-9$

  $\Rightarrow 112={\left(11\right)}^2-{\left(3\right)}^2$

  $\Rightarrow 112=\left(11-3\right)\left(11+3\right)$

  $\Rightarrow 112=\left(11-3\right)\left(11+3\right)$

  $\Rightarrow 112=\left(8\right)\left(14\right)$

Thus, the factors of 112 are 8 and 14.