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.

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

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.

We can summarize our process in the following way:

Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. This type of polynomial is unique because it can be factored into two binomials but has only two terms.

You will want to become familiar with the special relationship between a difference of squares and its factorization as we can use this equation to factor any differences of squares.

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.  Let’s look at an example of difference of squares to help us understand how this works.  We will start from the product of two binomials to see the pattern.

Given the product of two binomials: [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 each term can be written as something squared.  A difference of squares will always factor in the following way:

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:

[latex]2[/latex], and [latex]-2[/latex] have a sum of [latex]0[/latex]. You can use these to factor [latex]x^{2}–4[/latex].

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].

Now that we have seen how to factor a difference of squares with regrouping, let’s try some examples using the short-cut.

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.

TIP:  To help identify difference of squares factoring problems, make a list of perfect squares and become familiar with these values.  You will frequently see them in these types of factoring problems.  In addition, these problems must have a minus sign. For example, x^2 – 9 = (x+3)(x-3)

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

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