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.

In our first example, we solve the problem two different ways to show how to use the above formulas while also demonstrating that this strategy will produce the same result as the techniques we learned earlier for factoring trinomials of this form.

In the previous example, note that the results do match since [latex](x-7)(x-7)=(x-7)^2[/latex]. Which way did you prefer? You may have found it easier to simply use the Product and Sum Strategy. However, as the problems become more complicated, the benefits of identifying Perfect Square Trinomials become more apparent.

The next example would have required AC-Method, where instead, we will apply our new formula.

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 [latex]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 formula.

It is worth noting that these problems could also be factored using the techniques detailed in Section 6.3 (if [latex]a=1[/latex]) and Section 6.4 (if [latex]a\neq 1[/latex]). However, if we recognize that a trinomial is in the form of a perfect square trinomial, this process can serve as a nice shortcut.

We can summarize our process in the following way:

Let us look at one more example. In this one, notice that there are now two variables.

Difference of Squares

We would be remiss if we failed to introduce one more type of polynomial that can be factored. This polynomial can be factored into two binomials but has only two terms.  Let’s start from the product of two special binomials to see the pattern.

Consider the product of the following two binomials, which are identical except for the operation: [latex]\left(x-2\right)\left(x+2\right)[/latex]. If we multiply them together, we lose the middle term that we are used to seeing as a result.

Multiply:

[latex]\begin{array}{l}\left(x-2\right)\left(x+2\right)\\\text{}\\=x^2-2x+2x-2^2\\\text{}\\=x^2-2^2\\\text{}\\=x^2-4\end{array}[/latex]

The polynomial [latex]x^2-4[/latex] is called a difference of squares because teach term is a perfect square and the operation connecting them is subtraction.  A difference of squares will always factor in the following way:

To further verify this, it is interesting to note that we could also use the techniques of the previous sections. Let’s factor [latex]x^{2}–4[/latex] by writing it as a trinomial, [latex]x^{2}+0x–4[/latex].  This is similar in format to the trinomials we have been factoring so far, so let’s use the same method.

Find the factors of [latex]a\cdot{c}[/latex] whose sum is b, in this case, 0:

2 and -2 have a sum of 0. You can use these to factor [latex]x^{2}–4[/latex]. Using the Product and Sum Method in Section 6.3, results in [latex]x^2-4=(x+2)(x-2)[/latex].

However, applying the formula given above gives us an even quicker shortcut! Let us explore this same example one more time using our difference of squares equation.

Since order doesn’t matter with multiplication, the answer can also be written as [latex]\left(x-2\right)\left(x+2\right)[/latex].

You can check the answer by multiplying [latex]\left(x+2\right)\left(x-2\right)=x^{2}+2x–2x–4=x^{2}–4[/latex].

The following video show two more examples of factoring a difference of squares.

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

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.

And as we have seen in the previous sections, we must always be on the lookout for first factoring out a GCF.

As we have described, there are two key components to a difference of squares, perfect squares and the difference. We explore the importance of the operation further in the “Think About It” below.

Summary

In all factoring problems, we first look for a common factor that can be factored out of all terms. After achieving this (if possible), we have now learned two more factoring techniques in this section.

If a trinomial is of the form of a perfect square trinomial, it factors as [latex]a^2+2ab+b^2=(a+b)^2[/latex] or [latex]a^2-2ab+b^2=(a-b)^2[/latex].

If a binomial is of the form of a difference of squares, [latex]a^2-b^2[/latex], it will factor as [latex]a^2-b^2=\left(a+b\right)\left(a-b\right)[/latex].