Factoring Perfect Square Trinomials and Difference of Squares Binomials

Learning Objectives

  • Special Cases – Squares
    • Factor a polynomial of the form: [latex]{a}^{2}+2ab+{b}^{2}[/latex]
    • Factor a polynomial of the form: [latex]{a}^{2}-{b}^{2}[/latex]

Why learn how to factor special cases?

Repeated pattern of interlocking plus signs, each row a different color following the rainbow spectrum.

 

Some people like to find patterns in the world around them, like a game.  There are some polynomials that, when factored, follow a specific pattern.

 

Perfect square trinomials of the form: [latex]{a}^{2}+2ab+{b}^{2}[/latex]

A difference of squares: [latex]{a}^{2}-{b}^{2}[/latex]

 

In this lesson you will see you can factor each of these types of polynomials following a specific pattern.

Some people find it helpful to know when they can take a shortcut to avoid doing extra work.  There are some polynomials that will always factor a certain way, and for those we offer a shortcut.  Most people find it helpful to memorize the factored form of a perfect square trinomial or a difference of squares.  The most important skill you will use in this section will be recognizing when you can use the shortcuts.

Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

[latex]\begin{array}{ccc}\hfill {a}^{2}+2ab+{b}^{2}& =& {\left(a+b\right)}^{2}\hfill \\ & \text{and}& \\ \hfill {a}^{2}-2ab+{b}^{2}& =& {\left(a-b\right)}^{2}\hfill \end{array}[/latex]

We can use this equation to factor any perfect square trinomial.

A General Note: Perfect Square Trinomials

A perfect square trinomial can be written as the square of a binomial:

[latex]{a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}[/latex]

 

In the following example we will show you how to define a, and b so you can use the shortcut.

Exercises

Factor [latex]25{x}^{2}+20x+4[/latex].

In the next example, we will show that we can use [latex]1 = 1^2[/latex] to factor a polynomial with a term equal to 1.

Example

Factor [latex]49{x}^{2}-14x+1[/latex].

In the following video we provide another short description of what a perfect square trinomial is, and show how to factor them using a the formula.

We can summarize our process in the following way:

Given a perfect square trinomial, factor it into the square of a binomial.

  1. Confirm that the first and last term are perfect squares.
  2. Confirm that the middle term is twice the product of [latex]ab[/latex].
  3. Write the factored form as [latex]{\left(a+b\right)}^{2}[/latex], or[latex]{\left(a-b\right)}^{2}[/latex].

 

Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]

We can use this equation to factor any differences of squares.

A General Note: Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

[latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex]

Example

Factor [latex]9{x}^{2}-25[/latex].

The most helpful thing for recognizing a difference of squares that can be factored with the shortcut is knowing which numbers are perfect squares, as you will see in the next example.

Example

Factor [latex]81{y}^{2}-144[/latex].

 

In the following video we show another example of how to use the formula for fact a difference of squares.

We can summarize the process for factoring a difference of squares with the shortcut this way:

How To: Given a difference of squares, factor it into binomials.

  1. Confirm that the first and last term are perfect squares.
  2. Write the factored form as [latex]\left(a+b\right)\left(a-b\right)[/latex].

Think About It

Is there a formula to factor the sum of squares, [latex]a^2+b^2[/latex], into a product of two binomials?

Write down some ideas for how you would answer this in the box below before you look at the answer.

 

Factor Completely

Sometimes you may encounter a polynomial that takes an extra step to factor. In our next example we will first find the GCF of a trinomial, and after factoring it out we will be able to factor again so that we end up with a product of a monomial, and two binomials.

Example

Factor completely [latex]6m^2k-3mk-3k[/latex].

In our last example we show that it is important to factor out a GCF if there is one before you being using the techniques shown in this module.

Summary

In this section we used factoring with special cases. The last topic we covered was what it means to factor completely.