Advanced Difference of Squares

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Algebra One $\longrightarrow$
  • Confirm the knowledge we already have about the Difference of Squares factoring technique
  • Take the square root of multivariable monomials to use the Difference of Squares factoring method
  • Understand complex forms of a Difference of Squares such as perfect square trinomials
Lesson Description

Most of the time when we need to use Difference of Squares, it will be the cases we already looked at in the first Difference of Squares lesson. However, occasionally we need to factor slightly more complicated variable expressions using the Difference of Squares technique.

Practice Problems

Practice problems and worksheet coming soon!


What We Already Know

Recall that a Difference of Squares is an object of the form$$a^2 - b^2$$Most of the time when we have need to work with and factor Differences of Squares, we will be looking at ones that have $a$ and $b$ each equal to a pure number or a single variable raised to a power, like we saw in the factoring lesson » on factoring a Difference of Squares. For example, in that lesson we saw things like
$$x^2 - 25 = (x+5)(x-5)$$
$$49-y^2 = (7+y)(7-y)$$
$$r^4 - 81 = \big(r^2 + 9\big) \big(r^2 - 9\big)$$$$=\big(r^2 + 9\big) (r+3)(r-3)$$If you became an expert at recognizing and factoring these forms, you're already in stellar shape for the future. It's only once in a while that we'll be asked to understand and be able to factor a Difference of Squares that has multivariable components.
You Should Know
The single-variable and single-number Differences of Squares that we've already learned are nearly all we'll ever need because they are more useful. Those types of objects are more likely to factor into linear terms that will interact or cancel with other linear terms in a large algebra problem, such as factors from quadratics.

More Complex Forms

Any expression of the form $a^2 - b^2$ is a Difference of Squares, whether it is one of the simpler ones we've seen up to this point or one of the multivariable ones we're about to see. The way you are supposed to check and recognize any potential Difference of Squares is to first see if it looks like the $a^2 - b^2$ form, and then check to see that each term has a perfect square root.This is more or less how we proceeded in the original lesson, but we are now much better equipped to try and take the square root of any monomial, even ones that have multiple variables, thanks to our recent lesson on Taking Square Roots of Variable Expressions ». Let's take a look at one.
Example 1Factor.$$36x^4y^2 - 100w^{10}z^8$$$\blacktriangleright$ Check to see if each term has a perfect square root:$$\sqrt{36x^4y^2} \longrightarrow 6x^2 y$$$$\sqrt{100w^{10}z^8} \longrightarrow 10w^5 z^4Therefore,$$36x^4y^2 - 100w^{10}z^8 = \left( 6x^2 y + 10w^5 z^4 \right) \left( 6x^2 y - 10w^5 z^4 \right)$$
You Should Know
The only reason we couldn't look at these more complicated Diff of Squares forms back in the chapter on factoring is that we weren't yet equipped with the know-how of how to effectively take the square root of more complicated expressions. Now that we have mastered Taking Square Roots of Variable Expressions » we can work on these types of forms by thinking in terms of actually taking square roots.E.g. we should now be able to quickly compute things like $\sqrt{64x^6 y^{10}} = 8x^3 y^5$.
Another case where this is uncommon but something you may occasionally be asked to understand involves perfect square trinomials.
Example 2Factor.$$x^2 - 2xy + y^2 - 49$$The tip-off here is that we have some $x$ and $y$ business and then a suspicious lonely perfect square integer subtracted from it. The first three terms are a perfect square trinomial. Factoring it yields$$(x-y)^2 - 49$$Therefore, this object really is a Difference of Two Squares. We can factor it from here using the Difference of Squares factoring form.$$[(x-y)+7][(x-y)-7]$$

Forced Differences of Squares

You can actually force any difference of two objects to factor by using their square roots. This is uncommon, and sometimes causes extraneous solution issues, but in the right circumstance it can be helpful.$$x^2 - 3 \longrightarrow (x + \sqrt{3})(x - \sqrt{3})$$
We typically don't like square roots as much as whole numbers, so pulling this kind of trick out is not common. There are indeed a few very necessary times when we need to use this way of thinking, but it's only in situations where we need to get something to cancel no matter what. This is often useful when studying limits » of the form $0 \div 0$.
Example 3Simplify.$$\frac{2x^2-14}{x+\sqrt{7}}$$$\blacktriangleright$ Typically, when presented with the instructions "simplify" (or any instructions until you hit Calculus), your teacher expects you never to turn in an answer that has square roots in the denominator. One way to deal with this problem is to rationalize » the denominator, though this is a relatively odd form. However, this particular problem is a two-step cakewalk if we create a "forced" Difference of Squares in the numerator.$$\frac{2\big(x^2-7\big)}{x+\sqrt{7}}$$$$\frac{2(x+\sqrt{7})(x-\sqrt{7})}{x+\sqrt{7}}$$$$2(x-\sqrt{7})$$

Put It To The Test

Example 4Factor completely.$$625a^{12} b^8 -81x^4 y^8$$
Show solution
$$\blacktriangleright \,\, \left(25 a^6 b^4 + 9x^2 y^4 \right)\left(25 a^6 b^4 - 9x^2 y^4 \right)$$$$= \left(25 a^6 b^4 + 9x^2 y^4 \right) \left(5a^3 b^2 + 3x y^2 \right) \left(5a^3 b^2 - 3x y^2 \right)$$Make sure you go "all the way", otherwise expect to lose some credit.
Example 5Factor completely.$$2x^4 - 4x^2 y + y^2 - 36$$
Show solution
$\blacktriangleright$ As we saw in Example 2, there is reason to make a Difference of Squares our number one suspect when we are told to factor and we are given some trinomial with a perfect square integer tacked on the end.$$\left(2x^2-y\right)^2 - 36$$$$\left( 2x^2 - y + 6 \right) \left( 2x^2 - y - 6 \right)$$
Lesson Takeaways:
  • Know what to look for when it comes to Differences of Squares that have multiple variables
  • Utilize square root thought processes to check if any given difference qualifies as a Difference of Squares
  • Be able to recognize perfect square trinomials that your teachers give you as Difference of Squares problems
  • Understand, if not independently recognize, how a Difference of Squares can be "created by force" with any difference, though it has limited use

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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