Difference of Squares
Lesson Priority: VIP Knowledge
- Derive the Difference of Squares factoring form from what we know about FOIL
- Be able to use the factoring method forward and backward
- Utilize the Difference of Squares method when a GCF is present in both objects
- Utilize the Difference of Squares method when one of the squared objects is a perfect square trinomial
A binomial that consists of one squared object subtracted from another squared object is a very special form that factors into a product of two binomials. Beside classic trinomial factoring, this form, called the Difference of Squares, is one of the most popular forms we encounter through our entire math career that we are expected to know how to factor.
Practice problems and worksheet coming soon!
A New Factoring Tool Note: this lesson sticks to monomials with one variable. If you are looking to practice using the Difference of Squares technique on monomials with two or more variables, head over to the lesson on Advanced Difference of Squares ».While most of the quadratics you'll be asked to factor in your math career will look similar to the trinomial form we saw in recent lessons, there are a couple special other forms that factor very quickly and neatly. By far, the most frequent and useful one is a quadratic form called Difference of Squares. We'll see this factoring technique almost as much as trinomial factoring, and we'll also see it used as a tool for other processes along our Algebra journey and beyond.So what exactly does this Difference of Squares look like? It actually looks just like its name: one squared object subtracted from another squared object. Here's a classic example:$$a^2 - b^2$$(1)
A Difference of Squares is a binomial object of the form $a^2 -b^2$, where $a$ and $b$ can be any monomial or constant.For example, here are some binomials that are and are not Differences of Squares.$$x^2-49$$This is a Difference of Squares, where $a=x$ and $b=7$, such that $a^2-b^2 = x^2 - 49$.$$w^4-z^2$$This is also one, where $a=w^2$ and $b=z$.$$x-6$$$x-6$ is not a Difference of Squares, as neither $x$ nor $6$ is a perfect square.$$9r^2-16$$This one is a Difference of Squares, where $a=3r$ and $b=4$.$$9r^2+16$$$9r^2+16$ is not a Difference of Squares, but rather a sum. Sums of Squares have an occasional use in advanced algebra, but we will not study them as part of our current chapter on Factoring, as they have no factoring forms in the realm of real numbers.These expressions look simple enough - so what's so special about them? As it turns out, every Difference of Squares expression can be factored the same way.
Any Difference of squares can be factored according to the following pattern:$$a^2-b^2=(a+b)(a-b)$$We saw this specific result when we first learned the FOIL Method, and while it's nice to know a little trick for using FOIL on things like $(a+b)(a-b)$ to quickly obtain $a^2-b^2$, it is even more important to recognize a Difference of Squares when we start with something like $a^2-b^2$ and we are asked to factor it.Here's a quick look at the derivation of the Theorem, using the FOIL method we previously learned.$$(a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2$$$$\therefore a^2-b^2 = (a+b)(a-b)$$We can see that when we multiplied $(a+b)(a-b)$ using FOIL that the Inner and Outer terms were $-ab$ and $+ab$, which are like terms that combine to zero and therefore disappear.Lets look at the few Difference of Squares expressions that we wrote down above:
Example 1$$x^2-49$$$\blacktriangleright$ Since $x^2$ and $49$ are each perfect squares, we will use their square roots in the factoring.$$a^2-b^2 = (a+b)(a-b)$$$$\Rightarrow x^2-49 = (x+7)(x-7)$$
Example 2$$w^4 -z^2$$$\blacktriangleright$ Again, figure out what the $a$ and $b$ values are which, when squared, give you $w^4$ and $z^2$, respectively.$$a^2-b^2 = (a+b)(a-b)$$$$\Rightarrow w^4-z^2 = \left(w^2+z\right)\left(w^2-z\right)$$
Example 3$$9r^2-16$$$\blacktriangleright$ Just like before, look for the square roots of each perfect square.$$9r^2 -16 = (3r+4)(3r-4)$$
It is helpful to focus on the concept here and not just the symbols in the formula. In words, when you have two squared objects where one is subtracted from the other, then to factor the Difference of Squares, take the square root of each, and multiply their sum and their difference.
Difference of Squares in PracticeLike any tool or method we see, and especially for one that we'll see over and over again, we want to make sure we know how it will be used, and how to do it consistently and correctly. Fortunately, factoring a Difference of Squares is a quick, structured process, so with just a little bit of practice you'll have it down in no time.The typical scenario that you will be presented with is to take the given Difference of Squares and re-write it in the factored form. Unlike many methods and topics in math, there is little-to-no scratch work or in-between steps. In fact, as adamant as Mr. Math is about doing methods step-by-step and writing out each step in your scratch work 100% of your math life, this is a rare exception! If you just "know" the answer for Difference of Squares problems, it is ok to simply write the answer.So let's get to it! Here's the deal for handling Difference of Squares factoring, step by step using the following example:
Example 4$$4t^2 - 25$$Step 1 - Factor out the GCF if one is presentI will always remind you of this during any type of factoring method - you'll make your life unnecessarily difficult in all factoring situations if you fail to recognize and factor out a Greatest Common Factor. We'll soon see some examples of this shortly, though we'll start with some examples without a GCF, like our above example of $4t^2-25$, which does not have a GCF to factor out.Step 2 - Verify that you have a Difference of SquaresAlthough you're here in "learning mode" for this topic, and will therefore do some rapid-fire back-to-back practice problems for this method, you'll find that often times when you need to use this method, it will be mixed in with other factoring problems. Therefore, this step is the simplest: verify that you have stumbled upon a Difference of Squares situation.In our above example, we can see that $4t^2-25$ does indeed fit the bill, since both $4t^2$ and $25$ are both perfect squares, and one is being subtracted from the other. If a GCF was factored out of the problem you're working on, then you would be examining the remaining factor, not the GCF.Step 3 - Compute the square root of each perfect squareSince our job is ultimately to look at a Difference of Squares and write the factored form, and since the factored form uses the square root of both items, we should identify the square root of each term. We can look again to our above example: when we looked at $4t^2-25$, we needed to determine that the square root of $4t^2$ is $2t$, and the square root of $25$ is $5$.Step 4 - Put it all togetherEach piece is then put in its place in our formula. Don't stress about memorizing the formula in symbols, because with just a little practice, this know-how will be firmly planted in your brain as a concept, not just random symbols.For our example, this final step is to take the $2t$ and the $5$ and place them in the formula. Multiply their sum and difference, which gives us$$4t^2-25 = (2t+5)(2t-5)$$
The Difference of Squares factoring method applies to any two monomials. Because of where this lesson lies in the DNA of Math, we will only work with single-variable monomials in this lesson. Later on, we will revisit the Difference of Squares factoring method and use more complicated monomials, in a lesson called Advanced Difference of Squares » .Let's look at one more example, this time with a problem that has a GCF.
$$2x^2-72$$$\blacktriangleright$ We'll step through this once again using the four step approach.Step 1There is a GCF between these two terms, so that's where we will begin.$$2x^2-72 = 2\left(x^2-36\right)$$It's only a constant, but it very important nonetheless. Without factoring out that $2$, we may not have recognized the remaining piece as a Difference of Squares. Since $x^2-36$ is a Difference of Squares, we will proceed with factoring it.Step 2All we will do in this step is determine that we do indeed have a Difference of Squares. As we just said a moment ago, $x^2 - 36$ is a Difference in Squares.Step 3Now we will look at the Difference of Squares $x^2 - 36$ and jot down each term's square root.$$x^2 \longrightarrow x$$$$36 \longrightarrow 6$$Step 4Finally, put together these results with the Difference of Squares factoring form:$$2x^2 - 72$$$$=2\left(x^2 - 36\right)$$$$=2(x+6)(x-6)$$Note that as you become more affluent and practiced, you will not be stepping through the four steps explicitly, but rather able to move through these problems with 2 or 3 steps of scratch work.
Always, always, always remove a Greatest Common Factor first when you are trying to do any type of factoring ever! Failure to do so will make your job ridiculously more difficult.
Put It To The TestLet's crunch through some examples and make sure we're ready to rock any time in the future that we need to factor using this technique.
$\blacktriangleright$ Move through the steps as needed: check for GCF and that you have a Difference of Squares, take the square root of each term, then put the pieces in their place.$$q^2 - 64r^2$$$$\longrightarrow a = q$$$$\longrightarrow b = 8r$$$$q^2 - 64r^2$$$$=(q+8r)(q-8r)$$
$$\blacktriangleright \,\, a = c^4$$$$\longrightarrow b = d^3$$$$c^8 - d^6$$$$=(c^4+d^3)(c^4-d^3)$$
$\blacktriangleright$ This example has a GCF that needs to be factored out, so that we can proceed more easily.$$3x^5-27x^3$$$$=3x^3\left( x^2 - 9\right)$$Now we can finish the problem.$$\longrightarrow a = x$$$$\longrightarrow b = 3$$$$3x^5-27x^3$$$$=3x^3 \left(x-9\right)$$$$=3x^3 (x+3)(x-3)$$
$\blacktriangleright$ The first part of this solution is not new to us by now:$$x^4-1 = \left(x^2+1\right)\left(x^2-1\right)$$However, if you stop there, many teachers will take points off. Of the two factors we obtained, the first is a Sum of Squares, which we can do nothing more with, but the second piece is another Difference of Squares, which we must factor further, particularly if our instructions are to "factor completely".$$\left(x^2-1\right) = (x+1)(x-1)$$$$\therefore x^4-1 =$$$$\left(x^2+1\right)(x+1)(x-1)$$
Nearly always, when asked to factor something, our teachers are shrewd enough to write instructions that say "factor completely". Regardless, most teachers will expect that if you are factoring using Difference of Squares, and one of the factors you obtain is itself a Difference of Squares, that you will proceed to factor that piece as well (and its pieces, if possible, and their pieces if possible, and so on).
- Understand and recognize what a Difference of Squares binomial looks like
- Be able to look at a binomial and quickly decide whether Difference of Squares factoring is applicable to it, regardless of the presence of a GCF
- Use the Difference of Squares factoring form consistently and correctly for perfect square variable expressions