Chapter 14.8, Problem 22HP

To determine

To calculate: The pattern that followed by the binomial $x^2-1,x^2-4,x^2-36,$ and so on.

Answer to Problem 22HP

The pattern that followed by all the binomial expressions is $a^2-b^2$ .

Explanation of Solution

Given Information:

The binomial expressionsis $x^2-1,x^2-4,x^2-36$ .

Calculation:

The given binomial expressions are $x^2-1,x^2-4,x^2-36$ and so on.

Consider the binomial expression $x^2-1$ .

Find the factors of the expression $x^2-1$ .

  $x^2-1=\left(x-1\right)\left(x+1\right)$

For the binomial expression $x^2-4$ .

Find the factors of the expression $x^2-4$ .

  $\begin{array}{c}{x}^2-4=x^2-2^2\\ =\left(x-2\right)\left(x+2\right)\end{array}$

Similarly, for the binomial expression $x^2-36$ .

Find the factors of the expression $x^2-36$ .

  $\begin{array}{c}{x}^2-36=x^2-6^2\\ =\left(x-6\right)\left(x+6\right)\end{array}$

From above calculation it is cleared that all the binomial expression follow the algebraic formula $a^2-b^2$ .

Factors of $a^2-b^2$ are:

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

Hence, the pattern that followed by all the binomial expressions is $a^2-b^2$ .