Algebra basics (ASL)
- Factoring perfect squares
- Factoring quadratics: Perfect squares (ASL)
- Identifying perfect square form
- Factoring higher-degree polynomials: Common factor
- Factoring perfect squares: negative common factor
- Factoring perfect squares: missing values
- Factoring perfect squares: shared factors
- Difference of squares intro
- Perfect squares
- Factoring quadratics in any form
Factoring quadratics: Perfect squares (ASL)
Learn how to factor quadratics that have the "perfect square" form. For example, write x²+6x+9 as (x+3)².
Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.
In this article, we'll learn how to factor perfect square trinomials using special patterns. This reverses the process of squaring a binomial, so you'll want to understand that completely before proceeding.
Intro: Factoring perfect square trinomials
To expand any binomial, we can apply one of the following patterns.
Note that in the patterns, and can be any algebraic expression. For example, suppose we want to expand . In this case, and , and so we get:
You can check this pattern by using multiplication to expand .
The reverse of this expansion process is a form of factoring. If we rewrite the equations in the reverse order, we will have patterns for factoring polynomials of the form .
We can apply the first pattern to factor . Here we have and .
Expressions of this form are called perfect square trinomials. The name reflects the fact that this type of three termed polynomial can be expressed as a perfect square!
Let's take a look at a few examples in which we factor perfect square trinomials using this pattern.
Example 1: Factoring
Notice that both the first and last terms are perfect squares: and . Additionally, notice that the middle term is two times the product of the numbers that are squared: .
This tells us that the polynomial is a perfect square trinomial, and so we can use the following factoring pattern.
In our case, and . We can factor our polynomial as follows:
We can check our work by expanding :
Check your understanding
1) Factor .
2) Factor .
3) Factor .
Example 2: Factoring
It is not necessary for the leading coefficient of a perfect square trinomial to be .
For example, in , notice that both the first and last terms are perfect squares: and . Additionally, notice that the middle term is two times the product of the numbers that are squared: .
Because it satisfies the above conditions, is also a perfect square trinomial. We can again apply the following factoring pattern.
In this case, and . The polynomial factors as follows:
We can check our work by expanding .
Check your understanding
4) Factor .
5) Factor .
6*) Factor .
7*) Factor .
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