Factoring - Difference of Two Squares
Transcript
This lesson discusses perhaps the single most important factoring pattern in all of algebra. We already learnt 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 are learning, is clearly the most important. I'll just say, if you're not familiar with the Square of a Sum and the Square of a Difference, we talked about those in the video unfoiling.
And, so if you haven't seen that video maybe helpful to watch that video 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- b squared and this factors into (a + b) (a- b). It's easy to see why this is true by using FOIL on the right side.
So if we just start with (a + b)(a- b) and then simply FOIL out. Well, product of the first term is a squared, product of the outer is a times -b, that would be -ab. Product of the inner, a times b, that would be ab and product of the last b times -b would be -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- 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- 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 + 7)(x- 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 9 x 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- 4)(3x + 4).
Incidentally notice, it doesn't matter at all whether we put 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 first or the one with 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 + 8y)( 5x- 8y).
Finally, x squared y squared- 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 1, so this will fact out to (xy- 1)(xy + 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 any even integer as the sum of two of the same integer. So of course, something like 10 would be 5 + 5, 18 = 9 + 9. It's very easy to express any integer as as the sum of two with the same integer. And so this is helpful when we think of this sum of powers. Now if you're not familiar with the sum of powers rule, this is something that we talked about in the multiplying expressions video.
A fewer video lessons back. So if you haven't been watching this video 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 skipped ahead to the power and roots module. But at the very least, if you've been watching the videos in order, you would have met this exponent rule at this point.
So go back and watch that if it's unfamiliar. But what this means is we could split any even exponent into two powers with equal exponents. So, for example, if we had x to the 10th, we can, of course, write 10 as 5 plus 5. And x to the 5 + 5, that would be x to the 5th times x to the 5th, so this is the sum of the exponents rule.
And, of course, x to the 5th times x to the 5th that means times itself, that is x to the 5th squared. And so this s 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 5th squared, x to the 5th 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 5th 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 the Power and 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 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 6th is x cubed squared. x to the 8th is x to the 4th squared and x to the 12th is x to the 6th squared. Since all even powers are squares, we can use the Difference of Two Squares to factor expressions involving even powers. So let's start out with x to the 6- 16. Well, clearly b is 4 because 4 squared is 16, well, x to the 6 is the square of x to the 3rd,.
So a would be x to the 3rd, so a would be x to the 3rd, so we would get (x to the 3rd- 4)(x to the 3rd + 4), that would be the factored form. If we have x to 8th- 9y squared, well, x to the 8th is x to the 4 squared. So a would be x to the 4, and, of course, be would be 3y. So this factors into (x to the fourth + 3y)(x to the fourth- 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 videos ago. Factoring out a greatest common factor, we factor out an x to the 5th. Then we're left with x squared- 4, so we've already written this as a product but we can factor it even further.
We can factor using the difference reduced square patterns and that is the fully factored form of that original binomial. Here's another one. So, of course, we can factor this, x to the 4th is x squared, squared so a equals x squared, b equals 9. We can factor this into (x squared + 9)(x squared- 9), but notice that x squared- 9 is another difference of squares.
So we can factor that one again, And factor this, so we have (x squared + 9)(x- 3)( x + 3), and that is the fully factored form. So any difference of squares, we can factor further. Notice that x squared + 9 that's a sum of squares there's actually no way to factor that.
It's worth noting that there is 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 + b) squared = a squared + 2ab + b squared. The square of a difference (a- b) squared is a squared- 2ab + b squared. And then the really important one the difference of two squares (a + b) (a- b) is a squared- b squared.
So here is some practice problems, pause the video and factor these fully and then we will talk about them. First one is very easy to factor. Second one, we can factor this. The third one, we factor at an x and then we factor out into x to the 4th + 1, x to the 4th- 1, but that's a different of squares that second one, so we could factor that further.
And then that x squared- 1 is another difference is squared so we can factor that further, so it turns out we get 5 different factors that is the fully factored form of that last expression. Here's a practice problem that's little more test like, of course, there are only four choices instead of five choices, but this is a multiple choice test in the feel of what you might see on the real test.
So pause the video, and then well talk about this. Okay, so let's think about this. Lets start out with these first two,equations, y = 5 + x and y = 12- x. We rewrite a little bit, we see what we are given here is what y- x equals, and what y + x equals, so that's interesting hold on to that for a moment. Then we what to know k equals, well, clearly k = y squared- x squared, well, we know we can factor y squared- x squared into (y- x)(y + x), and we know those two values.
We know (y- x) is 5 and (y + x) is 12, so this is just 5*12, which is 60, so k = 60. In this lesson, we discussed the difference of two squares pattern and how to use it in factoring algebraic expressions.
Read full transcriptAnd, so if you haven't seen that video maybe helpful to watch that video 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- b squared and this factors into (a + b) (a- b). It's easy to see why this is true by using FOIL on the right side.
So if we just start with (a + b)(a- b) and then simply FOIL out. Well, product of the first term is a squared, product of the outer is a times -b, that would be -ab. Product of the inner, a times b, that would be ab and product of the last b times -b would be -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- 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- 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 + 7)(x- 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 9 x 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- 4)(3x + 4).
Incidentally notice, it doesn't matter at all whether we put 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 first or the one with 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 + 8y)( 5x- 8y).
Finally, x squared y squared- 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 1, so this will fact out to (xy- 1)(xy + 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 any even integer as the sum of two of the same integer. So of course, something like 10 would be 5 + 5, 18 = 9 + 9. It's very easy to express any integer as as the sum of two with the same integer. And so this is helpful when we think of this sum of powers. Now if you're not familiar with the sum of powers rule, this is something that we talked about in the multiplying expressions video.
A fewer video lessons back. So if you haven't been watching this video 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 skipped ahead to the power and roots module. But at the very least, if you've been watching the videos in order, you would have met this exponent rule at this point.
So go back and watch that if it's unfamiliar. But what this means is we could split any even exponent into two powers with equal exponents. So, for example, if we had x to the 10th, we can, of course, write 10 as 5 plus 5. And x to the 5 + 5, that would be x to the 5th times x to the 5th, so this is the sum of the exponents rule.
And, of course, x to the 5th times x to the 5th that means times itself, that is x to the 5th squared. And so this s 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 5th squared, x to the 5th 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 5th 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 the Power and 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 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 6th is x cubed squared. x to the 8th is x to the 4th squared and x to the 12th is x to the 6th squared. Since all even powers are squares, we can use the Difference of Two Squares to factor expressions involving even powers. So let's start out with x to the 6- 16. Well, clearly b is 4 because 4 squared is 16, well, x to the 6 is the square of x to the 3rd,.
So a would be x to the 3rd, so a would be x to the 3rd, so we would get (x to the 3rd- 4)(x to the 3rd + 4), that would be the factored form. If we have x to 8th- 9y squared, well, x to the 8th is x to the 4 squared. So a would be x to the 4, and, of course, be would be 3y. So this factors into (x to the fourth + 3y)(x to the fourth- 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 videos ago. Factoring out a greatest common factor, we factor out an x to the 5th. Then we're left with x squared- 4, so we've already written this as a product but we can factor it even further.
We can factor using the difference reduced square patterns and that is the fully factored form of that original binomial. Here's another one. So, of course, we can factor this, x to the 4th is x squared, squared so a equals x squared, b equals 9. We can factor this into (x squared + 9)(x squared- 9), but notice that x squared- 9 is another difference of squares.
So we can factor that one again, And factor this, so we have (x squared + 9)(x- 3)( x + 3), and that is the fully factored form. So any difference of squares, we can factor further. Notice that x squared + 9 that's a sum of squares there's actually no way to factor that.
It's worth noting that there is 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 + b) squared = a squared + 2ab + b squared. The square of a difference (a- b) squared is a squared- 2ab + b squared. And then the really important one the difference of two squares (a + b) (a- b) is a squared- b squared.
So here is some practice problems, pause the video and factor these fully and then we will talk about them. First one is very easy to factor. Second one, we can factor this. The third one, we factor at an x and then we factor out into x to the 4th + 1, x to the 4th- 1, but that's a different of squares that second one, so we could factor that further.
And then that x squared- 1 is another difference is squared so we can factor that further, so it turns out we get 5 different factors that is the fully factored form of that last expression. Here's a practice problem that's little more test like, of course, there are only four choices instead of five choices, but this is a multiple choice test in the feel of what you might see on the real test.
So pause the video, and then well talk about this. Okay, so let's think about this. Lets start out with these first two,equations, y = 5 + x and y = 12- x. We rewrite a little bit, we see what we are given here is what y- x equals, and what y + x equals, so that's interesting hold on to that for a moment. Then we what to know k equals, well, clearly k = y squared- x squared, well, we know we can factor y squared- x squared into (y- x)(y + x), and we know those two values.
We know (y- x) is 5 and (y + x) is 12, so this is just 5*12, which is 60, so k = 60. In this lesson, we discussed the difference of two squares pattern and how to use it in factoring algebraic expressions.
