Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College. The sum and difference of cubes. A binomial of the form a ^3 + b ^3 factors into the quantity A+B times the quantity a ^2 minus AB plus b ^2. The difference of cubes, the binomial a ^3 -, b cube, factors into the quantity a -, b times the quantity a ^2 plus AB plus b ^2. These formulas are very similar, and sometimes we talk about mopping or matching the sign of the original binomial, then doing the opposite. So if it's positive, it's going to be negative. If it was negative, it's going to be positive. And the last one is always a positive because it's actually going to be the term that's squared and a negative times a negative and a positive times a positive are square are both positive. So we might want to give ourselves a list of some of the common cubes. So over here on the right one cubed equal 1/2 cubed is 8, etcetera. I did up through 10, so X ^3 + 64. We could think of that as X quantity cubed +4 quantity cubed. So the easiest way I can tell you to do it is to think about the first parenthesis. Just drop the cubes IEX plus four. The next parenthesis take the first term and square it. Take the two terms multiplied together with the opposite sign, and then the last term and square it. And that's what X ^3 + 64 factors into. Let's do quite a few more examples. If we look at 64, that once again is 4 ^3. 125 X cubed is going to be 5X times 5X times 5X. So the first parenthesis, think about just dropping the cubes, then taking that first term and squaring it, multiplying the two together and changing the sign and the last 1 ^2. So 64 -, 125 X cubed factors into the quantity 4 -, 5 X times the quantity 16 plus twenty X + 25 X squared. What if we had a fraction? It's not any big deal, we're just going to treat it like it's a cube. What's 1 eighth cubed? Well, do the top. The numerator that's 1 * 1 * 1 gives US1, and do the bottom denominator 2 * 2 * 2 is going to give us 8. So 1/2 * 1/2 * 1/2 would give us 1/8. When we look at this, the first parenthesis drop those cubes, the second parenthesis, the first term squared, the two multiplied together with the opposite sign, and the last 1 ^2. So a ^3 + 1 eighth factors into the quantity A + 1/2 times the quantity a ^2 -, 1/2 A plus 1/4. This next one. We always have to take out anything they have in common first. This one has a 2Y in common, which would leave us y ^3 -, 64. Now that Y cube -64 inside the parentheses is the difference of cubes, so remember to bring the two Y along. But now we're going to have y ^3 to get the Y ^3 and four cubed to get the 64. When we factor this, the first parentheses is just drop those cubes, the second the first term squared, the two multiplied together with the opposite sign, and the last 1 ^2. Well, what if we get things with higher powers and that's still fine. Think about what times, what times? What gives me a cube? Well, the rule is a power to power. We multiply. So t ^2 ^3 would give me T to the 6th, six y ^2 ^3 would give me 216 Y to the 6th. So the first parenthesis, we're going to drop those cubes. The next parenthesis, the first term squared t ^2 ^2 powers to powers we multiply. So to the 4th, going to then multiply the two inner outer terms and change the sign. And then we're going to take that last term and square it. So T to the 6th plus 216Y to the 6th factors into t ^2 + 6 Y squared times the quantity T to the 4th minus six y ^2, t ^2 + 36 Y to the 4th. OK, couple more and then we'll go into some really challenging ones. If we look at this next one, we're going to factor out of two and then we want to think about what times, what times what gives us X to the 3A and what times, what times what would give us 8Y to the 3B. So we're going to have two X to the A cubed because powers to powers, we multiply A * 3 is the 3A plus the quantity 2 yb cubed. When we factor this, the first parenthesis, we drop those cubes. The second parenthesis, the first one squared, the two multiplied together, changing the sign and the last 1 ^2. Now this next one's meant to be tricky, but it's really not if we just follow the rules. If we drop our cubes, we'd get X + 5 + y -, 4. Then we're going to take that first one and square it, and in a minute you're going to see that we need to foil that out. The next two terms are multiply, or the next term is the 2 multiplied together, changing the sign. So instead of a plus, it's a minus. And then finally it's the last 1 ^2. Well, we need to do some simplifying here. So in this first parenthesis, we're going to get X + y and the +5 and the -5 are going to cancel. In the next parenthesis, this X + 5 ^2 is going to give us X ^2 + 10, X plus 25. Then if we look at these next two terms multiplied, we're going to have a minus, and let's put it in parenthesis. So we know that we're going to subtract everything we get XY minus 5X plus five y -, 25, and then finally we're going to have plus and we're going to take that y -, 5 and square it. So y ^2 -, 10 Y plus 25. When we combine our like terms in that big long parenthesis, we're always going to bring down the X + y, but now we're going to have an X ^2. If we look here, we have a 10X, but we also have a negative of a -5. XA negative of a negative is a positive, so we're going to get plus 15X. Then if we look, we have AY squared and we have a negative of +5 Y to combine with -10 Y. So we're going to get -15 Y. We also have a negative XY. And then we have our constants. We have a +20 five a -, a negative, which is also a +25 plus another 25 S plus 75. If we look, we've used up all of those terms. We've listed them all. That's what that would factor into. The next ones are going to involve some factoring that we need to take out first. So 116th is not a perfect square, but we could factor out. And I'm going to show this two different ways. We're going to, we could factor out a 116th. If we factor out a 116th, this first term would just be left X to the three. A 116th though times what would give me one half. Well, 1/2 is really just 8 sixteenths. So if I took out a 116th, that would leave me an 8. Why did the six AZ to the 9B? And then we can put that as the cube formula. We'd have our X to the a ^3 + 2 Y to the two AZ to the 3B, all cubed because powers to powers. Once again, we multiply. So now if we dropped the cubes, our first parenthesis would say X to the A2Y to the two, AZ to the 3B, our second parenthesis. That first term squared, the two multiplied together, changing the sign, and then the last term squared. That would give us an answer. Now we could have factored out 1/2. Instead, if we had pulled out a half, 1/2 * 1/8 would give me 116th, and then there wouldn't be a coefficient other than A1 on the second term in that parenthesis. So now we'd bring down the half. This is a different way to do it. I'm going to have two equivalent answers. We'd have 1/2 X to the a ^3 + y to the two, BZ to the 3B cubed. So when we do this one, we start by dropping those cubes just like we have in all the other examples, and then we would take that first one and square it. The two multiplied together, changing the sign and then the last 1 ^2. Those are equivalent answers and they are both correct. Except somewhere I changed that Y to the B. It was really 6A originally. So we change all those YS to A's in the power. If you look here, they really are equivalent answers, they just have different coefficients. But if we multiplied through, we'd end up with the same thing. So let's look at the last one. The last one we could factor out of 5/8 and we'd get 8X cubed Y to the 6th -1 or we could just factor out of five and get X ^3 y to the 6th -1 eighth 5/8. We'd end up with two XY squared -1 if we square that first term, four X ^2 y to the 4th, multiply the two together, changing the signs, and square the last term. Or if we had used this other method, we could have done XY to the XY squared minus 1/2 * X ^2, y to the 4th plus 1/2, XY squared plus 1/4. Those both will work. Thank you and have a wonderful day.