Factoring Difference of Two Perfect Squares
Factoring algebraic expressions are very important during your course of study to learn the basics of algebra. There are several different types of factoring cases, including using the greatest common factor (GCF), or using the grouping method, and also using special patterns where the factoring is very easy.
Among these special patterns, we find the difference of two perfect squares. By perfect squares, we mean two numbers raised to the power of 2.
When we have an algebraic expression written as the difference of two perfect squares, we've guaranteed the factoring using the easiest way possible. So how do we factor that difference?
Factoring Difference of Two Perfect Squares Formula
They are very familiar formulas, which can be simplified by the following scheme:
Now, let's move on to using the expression with variables:
As a note, when we say that and are real numbers, this means that they can be known, or we can express them or one of them with another unknown, or one or both of them can be composite numbers.
For example:
Examples of How to Factor Difference of Two Perfect Squares
Here is a set of illustrative examples of how to factor the difference of two perfect squares. And we will guide you through all the steps, you can skip them when you are skilled enough and get the result straight away.
Example1:
Factor the binomial below using the difference of two squares method.
Solution:
As for the first term, it is clear that it is written in the form of a power raised to the exponent 2, but the second term which is 25 is not, so here we must look for a number if we raise it to the power of 2, we get 25, no doubt that you knew it quite easily because it is a very simple case which is . So we will substitute by into the expression , we get:
We will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows:
Example2:
Factor the binomial below using the difference of two squares method.
Solution:
Both terms are not raised to the power of 2, here we must always show an exponent to be able to factor.
The number 81 is a perfect square of 9, i.e. .
But the number is a product of two numbers, using power rules, we rewrite it as a perfect square: .
Therefore,
Same as the example 1, we will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows:
Example3:
Factor the binomial below using the difference of two squares method.
Solution:
Both terms are not raised to the power of 2, here we must always show an exponent to be able to factor.
The number is a product of two numbers, using power rules, we rewrite it as a perfect square: .
And the number is a product of two numbers, using power rules, we rewrite it as a perfect square: .
Therefore,
Same as the examples above, we will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows:
Example4:
Factor the binomial below using the difference of two squares method.
Solution:
Both terms are not raised to the power of 2, here we must always show an exponent to be able to factor.
The number is a perfect square of 7, i.e, .
And the number is raised to the power of 2.
Therefore,
Same as the examples above, we will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows:
Note: You must simplify the expressions inside the parentheses by removing nested parentheses then applying combine like terms.
Example5:
Factor the binomial below using the difference of two squares method.
Solution:
Both terms of the expression are raised to the power of 4, wich mean that we can rewrite it in order to obtain the exponent 2. Note that all powers that the exponent is even we can rewrite it as power of 2.
Therefore,
Same as the examples above, we will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows:
You may think we are done the factoring, no, my friend, notice carefully the last pair of brackets, does it remind you of the expression ? Yes, it is a difference of two perfect squares, which means that it is factorable. Therefore,
Hence,
Example6:
Factor the binomial below using the difference of two squares method.
Solution:
As for the first term, it is clear that it is written in the form of a power raised to the exponent 2, but the second term which is 3 is not, so here we must look for a number if we raise it to the power of 2, we get 3. What is the number multiplied by itself to get 3? I hope you know it: . So we will substitute by into the expression , we get:
This formula seems familiar to you, yes it is exactly what we wrote in the previous rule. So we will take the base powers and , i.e. we take and , and then put them inside two pairs of parentheses, where there will be two opposite signs between the two numbers as follows: