Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Sum and difference of cubes. If I have something of the form a ^3 + b ^3, it's going to factor into just the ( A + B * a ^2 minus AB plus b ^2. And if I have a ^3 -, b ^3, that's going to factor into A minus BA squared plus AB plus b ^2. Sometimes we think about matching the signs, the opposite sign and positive or mop so match opposite and positive. So X ^3 + 64. We could think of what cubed gives me X ^3 and that's X and what cubed gives me 64 and that's four so X ^3 + 4 ^3. So now when I put it into the formula I get X + 4 * X ^2, multiply these two together and switch the sine and then take the last one and square it. Now to check this we would just multiply it out. So X * X ^2 - 4 X plus 16 + 4 * X ^2 - 4 X plus 16. When I do this, I get X ^3 -, 4 X squared plus 16X plus four X ^2 -, 16 X plus 64, and we can see the -4 X squared and +4 X squared cancel the -16 and +16. And we do indeed come up with what we started with. So for this next 164 that's going to be 4 ^3 again -125 is going to be 125 X cubed is going to be 5X cubed. So 4 - 5 X times 16 + 20 X plus 25 X squared. There's a pattern here. The first one squared, the two multiplied together, changing the sign, and then the last 1 ^2. This next one a ^3 + 1 eighth. Well, a quantity cubed. The 1/8 fractions sometimes upset people, but it's really actually pretty easy. Think about what cubed gives me a one, and then think about what cubed gives me an 8. So we're going to have that second parenthesis being 1/2. So using the formula, we get a + 1/2 * a -, 1/2, a plus 1/4, and I get the 1/4 by 1 * 1 on the top and 2 * 2 on the bottom. And that's my factorization. Now this next one, we need to realize that there's a 2Y that's in common before we do anything else. So we get AY cubed -64 so 2Y. And then we could think of this as y ^3 -, 4 ^3. So the two Y keeps coming along. And then we have y -, 4 Y squared plus four y + 16 T to the six. Well what cubed gives me T to the six and the answer is going to be t ^2 because with exponents powers to powers we multiply plus six y ^2 ^3. 6 ^3 is 216 powers to powers on the Y is we multiply. So to factor this we get t ^2 plus six y ^2 * t ^2 * t ^2 is T to the 4th. Multiply those two terms and change the sign. We usually write the variables in alphabetical order. T comes before Y and then the last one squared SO36Y to the 4th. This next one, we're going to start by pulling out A2 and we get X to the three a + 8 * y to the three B. So 2X to the what gives me X to the 3A when I cube it. And the answer is going to be X to the A because we multiply when it's power to power. The second one's going to be two yb cubed. So the two the first term X to the A + 2 yb, then we're going to square it X to the A * X to the A, or X to the a ^2 is X to the two A, multiply them together, changing the sign, and then the last 1 ^2. And that is our factorization. Really, it's just a formula X + 5 ^3 + y -, 5 ^3. So that's already a cube plus a cube. So we're going to have X + 5 + y -, 5. All of that's just our first parenthesis. Then we're going to have the X + 5 ^2, changing the sign from that plus. So it's going to be a -, X + 5 * y -, 5 plus that last 1 ^2. So there's going to be a lot of multiplication and combining like terms here. First thing, the X + 5 + y - 5. The +5 and -5 in this first parenthesis is just going to leave us X + y. This next one we have to foil things out, so we're going to get X ^2 + 10 X plus 25 minus. I'm going to actually leave it in parentheses for just a second. XY minus 5X plus five y -. 25 went ahead and left it in parentheses once I foiled it because of the negative in front, I personally sometimes lose my negative, so I'm just going to add myself an extra step so that I don't lose anything. This last one when I foil y ^2 -, 10 Y plus 25. So now that X + y was the first term, now we're just going to combine like terms. So I get an X ^2 and I get AY squared. Down here we usually put all of our squareds first. Then I get a negative XY. If I have +10 X and a negative -5 XI get positive 15X. And then the YS, I have a -5 Y here and a -10 Y here. So -15 Y and then I have a +25 and a negative of a -25, which is another positive and then another positive. So 75 total. And if I look, I've used up all of my terms. I could double check. There's my X ^2, there was my y ^2, there was my XY. Here's my X's with the other X that I combine like terms. Here's my YS with my other Y's that I combine like terms. And then finally, my three constants that I also combine now this next one with fractions. Sometimes it's easiest to think about getting a common denominator to see what's happening before we do much of anything. So if I have 116th X to the 3A and eight sixteenths Y to the six AZ to the 9B, I can see that the 16 is in common. So what if I factored that out? What if I pulled that outside? So I'm going to have 116th now, X to the three A + 8 Y to the six AZ to the 9B. Now that looks more like our formula. So I'm going to do 116th X to the a ^3, +2 Y to the two AZ to the 3B cubed. This is very similar to the one we did up here. So from here, actually, Nope, it's not quite exactly the same. So then we're going to have 116th X to the A + 2 Y to the two AZ to the 3B times X to the two A -, 2 X to the AY to the two AZ to the three B + 4 Y to the four AZ to the 6B. Remember, all I'm really doing is the formula X to the A * X to the A was the X to the 2A. These two terms multiplied together, changing the sign, and I just wrote them in alphabetic order. And then the last term squared 2 * 2, four Y to the two, a * y to the two, AY to the four, AZ to the three, b * Z to the three, BZ to the 6B. OK, so on this last example, we're going to realize that there's a 5 in common. So if I pull out A5, I get X ^3 y to the 6th -1 eighth. So then that XY squared quantity cubed. And remember the 1/8 is just 1 / 2 ^3 because one cubed is one and two cubed is 8. So we get five times XY squared minus 1/2 times. Multiply the first one by itself, so X ^2 y to the 4th. Multiply the inner terms and change the sign 1/2 XY squared. And then the last 1 ^2. 1 * 1 is 1/2 times two is 4. So that's our final answer. We've given you quite a few to try back here because we know that these can be tricky. Thank you and have a wonderful day.