How to Factor the Difference between Two Perfect Squares?

A step-by-step guide to factoring the difference between two perfect squares

Factoring the difference between two squares is a special case of factoring a polynomial, where you factor a binomial which is the difference between two terms that are both perfect squares.

Writing a binomial as the difference between two squares simply means that you rewrite a binomial as the product of two sets of parentheses. For example, \(a^2−b^2 =(a+b)(a−b)\). The binomial \(a^2−b^2\) can be factored into two sets of parentheses multiplied by each other. \((a+b)(a−b)\) will produce \(a^2−b^2\) when multiplied.

Not all terms can be factored in using this method. There are several clues to look for when determining whether an expression can be factored using the difference of squares. In the previous example a2-b2, each term is a perfect square and there is a subtraction symbol between each term. These are two useful clues to look for when determining whether a binomial can be factored using the difference between two squares. If these two clues exist, the expression can be factored using the difference of squares.

Factoring the Difference between Two Perfect Squares – Example 1:

Factor the following binomial using the difference of two squares method. \((x^2-81)\)

Solution:

The expression \((x^2-81)\) can be factored using the difference of two squares because both expressions are perfect squares and the expressions are separated by a subtraction sign. \(x^2\) is \(x\) times \(x\) and \(81\) is \(9\) times \(9\). \((x^2-81)\) can be rewritten as the product of \((x+9)(x-9)\).

Exercises for Factoring the Difference between Two Perfect Squares

Factor each polynomial using the difference of two squares method.

1. \(\color{blue}{(64-x^2)}\)
2. \(\color{blue}{(49x^2-25y^2)}\)
3. \(\color{blue}{(625\:m^4-36y^2)}\)
4. \(\color{blue}{(4x^4-144y^2)}\)

1. \(\color{blue}{(8+x)(8-x)}\)
2. \(\color{blue}{\left(7x+5y\right)\left(7x-5y\right)}\)
3. \(\color{blue}{\left(25m^2+6y\right)\left(25m^2-6y\right)}\)
4. \(\color{blue}{4\left(x^2+6y\right)\left(x^2-6y\right)}\)