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Factoring - Difference of Two Squares
This lesson discusses perhaps the single most important factoring pattern in all of algebra. We already learned about the square of a sum and the square of a difference. Those two with this new one are the big three factoring patterns, and this new one we’re learning is clearly the most important. I’ll just say, if you’re not familiar with the square of the sum and square of a difference, we talked about those in the video on foiling.
And so if you haven’t seen that video, it may be helpful to watch that video before we, before watching this particular video. This new pattern we’re learning is known as the difference of two squares. And as the name says, it’s just one square minus another square. So a squared minus b squared and this factors into a plus b times a minus b. It’s easy to see why this is true by using FOIL on the right side.
So we just start with a plus b times a minus b. And then simply FOIL out. Well, product of the first term is a squared. Product of the outer, is a times negative b, that would be negative ab. Product of the inner, a times b, that would be ab. And product of the last, b times negative b would be negative b squared.
Now miraculously those two cross terms. Negative ab and positive ab, those two cancel. And so, we’re just left with a squared minus b squared. So, this is an extremely elegant pattern, and this is precisely why the test is absolutely in love with this pattern. This pattern allows us to factor many different expressions.
So for example, starting out just with x squared minus 49. Well, obviously a would be x, because when we square it we get x squared and b would have to be 7, because when we square 7 we get 49 and so this is gonna factor into a plus b times a minus b, that would be x plus 7 times x minus 7. Here’s another one a little trickier because we have a coefficient of the x squared term.
9 is the square of 3 and so, 9x squared is 3x times 3x. So a equals 3x. And of course b equals 4. Because 4 squared is 16. And so we get 3x minus 4 times 3x plus 4. Incidentally notice, it doesn’t matter at all whether we put the min, the binomial with addition or subtraction first.
We’re multiplying these two. And of course multiplication is commutative. We can switch the order of the factors around. And so it really doesn’t make a difference at all. It is mathematically identical, whether we put the one with plus or the one minus first.
So you’ll notice in this video, I’ll just go back and forth, switching off between the two because they’re both equivalent. 25x squared minus 64y squared, the first one is 5x quantity squared, the second one is 8y quantity squared. So a equals 5x, b equals 8y, and this is 5x plus 8y times 5x minus 8y.
Finally, x squared y squared minus 1. X squared y squared, that would mean that a equals x times y, because x times y, times x times y would be x squared y squared, and of course b would be 1. This will factor into xy minus 1 times xy plus 1. The material covered by this pattern expands considerably, when we realize this.
Any even power of x, is the square of another power. So this is a big idea. First, we can express even, any even integer as the sum of two of the same integer. So of course something like 10 would be 5 plus 5. 18 equals 9 plus 9.
It’s very easy to express any integer as. As the sum of two of the same integer. And so, this is helpful when we think of the sum of powers. Now if you’re not familiar with this sum of powers rule, this is something that we’ve talked about in the multiplying expressions video, a few video lessons back.
So, if you haven’t been watching these videos in order, you might have missed that lesson. And that’s where we first talked about this exponent rule, and of course you can learn about all the exponent rules in depth, if you skip ahead to the power and roots module. But at the very least, if you’ve been watching these videos in order, you would’ve met this exponent rule at this point.
So go back and watch that if it’s unfamiliar. But what this means is we can split any even exponent into two powers with equal exponents. So for example, if we add x to the tenth we can of course write 10 as 5 plus 5, and x to the 5 plus 5 that would be x to the fifth times x to the fifth. So this is the sum of the exponents rule, and of course x to the 5th times x to the 5th, anything times itself, that is x to the 5th squared.
And so this is how we would write, an even power, as the square of something. I’ll also say, there’s another exponent rule that we haven’t talked about yet. But it may be familiar to you. This is the power to a power rule. So this is a little bit of a shortcut. If we have x to the fifth squared, x to the fifth squared means, that we multiply the exponents.
X to the 5 times 2. And of course we can write 10 as 5 times 2, so that means we can write it as x to the fifth squared. So this is another way to see the pattern. So again, if you’re familiar with this rule, you can think about it this way. If you’re not familiar with this rule, this is no problem.
We will study it in detail in Power Roots module. But you can just think about it the other way, having to do with addition. Either way, the even power of a variable is the square of a power with half the exponent. That’s the big idea here. And so, for example, here are a bunch of variables with even exponents.
And we can write each one as the square of the variable to half the exponent. So that means that x to the sixth is x cubed squared. X to the eighth is x to the fourth squared. And X to the 12th is x to the x to the sixth squared. Since all even powers are squares we can use the difference of two squares to factor expressions involving even powers.
So lets start out with x to the sixth. Minus 16. Well clearly b is, is 4, because 4 squared is 16. Well, x to the sixth is the square of x to the third. So, a would be x to the third. So we would get x to the third minus 4 times x to the third plus 4.
That would be the factored form. If we have x to the eighth minus 9y squared, well x to the eighth is x to the 4 squared. So a would be x to the four and of course b would be 3y. So this factors into x to the fourth plus 3y times x to the fourth minus 3y.
Okay this is interesting. We have odd exponents here. But notice the following. Notice we can factor out a greatest common factor. We talked about this a couple of videos ago, factoring out a greatest common factor, we factor out an x to the fifth,.
Then we’re left with x squared minus 4. So we’ve already written this as a product but we can factor even further. We can factor that x squared minus four, using the difference of two squared patterns. And that is the fully factored form, of that original binomial. Here’s another one.
So of course we can factor this. X of the fourth is x squared, squared. So a equals x squared, b equals 9. We can factor this into x squared plus 9 times x squared minus 9. But notice that x squared minus 9 is another difference of squares. So we can factor that one again.
And factor this so we have x squared plus 9, and times x minus 3 times x plus 3. And that is the fully factored form. So any difference of squares we can factor further. Notice that x squared plus 9, that’s a sum of squares. And there’s actually no way to factor that. It’s worth noting that there’s no way at all in algebra to factor the sum of two squares.
The big three patterns to know are. The square of a sum a plus b squared equals a squared plus 2ab plus b squared. The square of a difference, a minus b squared is a squared minus 2ab plus b squared and then the really important one, the difference of two squares. A plus b times a minus b is a squared minus b squared. So here’s some practice problems.
Pause the video and factor these fully, and then we’ll talk about them. First one is very easy to factor. Second one, we can factor this. Third one, we factor out an x, and then we factor out, into x to the fourth plus 1, x to the fourth minus 1, but that’s a difference of squares, that second one. So we can factor that further.
And then that x squared minus 1 is another difference of squared, so we can factor that further. So it turns out we get five different factors, that is the fully factored form of that last expression. Here’s a practice problem that’s a little more test like.
Of course, there are only four choices instead of five choices, but this is. A, a multiple choice test, in the, in the feel of what you might see on the real test. So pause the video, and then we’ll talk about this. Okay.
So let’s think about this. Let’s start out with these first two equations, y equals 5 plus x and y equals 12 minus x. We rewrite a little bit. We see what we’re given here is what y minus x equals and what y plus x equals. So that’s interesting, hold onto that for a moment.
Then we want to know what K equals. Well clearly K equals y squared minus x squared. Well we know we can factor y squared minus x squared. Into y minus x times y plus x, and we know those two values. We know y minus x is 5, and y plus x is 12, so this is just 5 times 12 which is 60.
So K equals 60. In this lesson, we discussed the Difference of Two Squares pattern, and how to use it in factoring algebraic expressions.
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