Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Multiplying, dividing rational expressions. We want to get a monomial over a monomial and then reduce. So this first example, we currently have monomials, so 7 / 21 seven goes into both 7 and 21, leaving us a one and a three, and a ^3 / a reduces to an A ^2. So we could think of this as just a ^2 / 3 or one a ^2 / 3. This next example, we're going to factor out of three in the denominator in order to get a monomial. A monomial being something that's just multiplied and or divided. Once we have a monomial in both the numerator and denominator at the top and the bottom, then we reduce. So 21 and three reduce to 7:00 and 1:00. So I could leave it as 7 / 1 * 2 X -3 or more traditionally it would be 7 / 2 X -3. The next example, we're going to factor out a three in the numerator, and we're going to factor out an X in the denominator. In this example, we can see that the X + 7 portions cancel, leaving us just a 3 / X sometimes. It's important to understand that factoring out a negative or a -1 just changes the signs on the inside of the parenthesis. So if I factor down a -1 in the numerator, that just turns it into five a -, 6. If I factor 2 out of the denominator, then I have a five a -, 6 again. So five a -, 6 / 5 A -6 would cancel, leaving us a negative 1/2 whenever we have parentheses in the top and the bottom that look exactly the same but the signs are opposites. Pull out a negative. If I put in an extra step or two. If I started just by realizing the top really didn't have anything in common and I pulled out A1, and then I realized the bottom had a 2. Here we can see that the numerator and the denominator look exactly the same except for a negative. And truthfully I could have pulled the negative even out of the bottom if I wanted to. But we don't typically use and leave a negative or use it in the denominator. So if I did that, the 6 -, 5 A's would cancel, but we'd still put the negative either up on top or way out in front. This next example is going to be difference of two squares on top, so 5 -, P and 5 + P and the bottom is a trinomial that's a square. So we have P + 5 * P + 5. Five plus P and P + 5 are really, really the same thing because we have the commutative property that says we can add in any order we want. So our final answer here is going to be 5 -, P / P + 5 or 5 + P either way. So a few for you to try. When we multiply and divide, it's going to be exactly what we've been doing. The only thing that's going to be added or another step is we're going to do 1 fraction than the other and then we're going to cancel in a multiplication. If something's on top and the exact same thing as anywhere in any of the terms on the bottom, they're going to cancel out. So for this first example, a ^2 - 1, we know that that's really just A + 1 * A - 1 because it's the difference of two squares. This 2 - 5 AI like my A's first, I find it a little easier to see, so I'm going to pull out a -1 and get five A -, 2 left. You don't have to do that. I just find it a little easier. The next term I'm going to pull a three out and get five, a -, 2. And finally this trinomial. I'm going to factor what are the two numbers that multiply to give me -6 but add to give me +5? Well, if they multiply to give me -6 I'm going to need a -1 and a +6. So a -, 1 A+ 6. Now literally if I find anything on top and bottom that are the same, I'm going to cancel. So a -, 1 up here, A -, 1 down here, then five, A -, 2 down here, five a -, 2 up there. So our final answer to this problem is going to be 3. We usually put the coefficient in front times a plus one over. We don't leave the negative in the denominator, so we usually put it way out in front. Or we can put it in front of just the three a + 6. The A's can't cancel because this a + 6 is all together one term. It's not a monomial. I can only cancel when everything's a monomial. Once everything's a monomial, then the top and the bottom have to be exactly the same to cancel for any parenthesis portion. Now the division is exactly the same as the multiplication, with the exception that we're going to change this ÷ to multiplication and flip the term that comes right after the ÷ Sometimes we think of it as skip, flip, and multiply. Just kind of a little rhyme to help you. So we skip the first term, we flip the second term, and then we multiply. So X ^3 + 8 Y cubed is the cube formula, and it factors to X + 2 yx minus X ^2 -, 2 XY plus four y ^2. The denominator factors into two ** +2 Y down here and plus AY, because when we think about our outside terms, we get 4 XY and the inside is one XY for a total of 5 XY. So now the next fraction, we're going to flip it. So I'm going to pull out A2 and I get a four X ^2 -, y ^2. Well, that's the difference of two squares. So that actually can go further. So that really turns into 2 * 2 X plus Y and two X -, y the denominator. We're going to factor out an X and we're going to see that that leaves us X ^2 -, 2 XY plus four y ^2. And that factors further into X. Oh no, that doesn't factor further because there aren't 2 numbers that multiply to me for and add to give me -2 So now we're going to start canceling. Here's an X + 2 Y on top and an X + 2 Y on bottom. Remember, inside the parentheses, they have to be exactly the same. Here's an X ^2 - 2 XY plus four y ^2. Here's an X ^2 - 2 XY plus four y ^2. Here is a 2X plus Y and A2 X + y. So once we cancel everything, then we just group together or combine what's left. So we have a 2 A2 X -, y all over an X. And that's our final answer. Couple for you to try. Now these last examples get a little more complex. So the first one we're going to have to distribute in order to combine like terms in order to factor and reduce. So we get X ^2 + X -, 2, X -6. The denominators are already a monomial, so I'm just going to leave the denominator alone. So the numerator is going to turn into X ^2 -, X -, 6 over. I'm going to just call it the common denominator for right now. Actually, let's just call it the denominator. Go ahead and write it out. So what 2 numbers are going to multiply to give me -6, but add to give me a -1, and the answer is going to be a positive 2 and a -3. So we're going to have X + 2 X -, 3 over that denominator and we can see the X plus twos will then cancel, leaving us a final solution of X -, 3 / X + 1 X +3. On this next example, we have difference of two squares in top, so we're going to have M minus TM plus T The bottom we're going to have to rearrange in order to factor because there's 5 terms, but m ^2 + 2 Mt plus t ^2 will factor. And then the m + t we're going to leave at the end for a minute. So when we look at these first 3 terms, we know that that factors into m + t quantity squared. And if we look at these last two terms, they don't have anything but a one in common. So we're going to factor out of one. So I went from 5 terms in the denominator to now 2 terms in the denominator. These two terms each have AM plus T in common. So if I factor out an m + t, that leaves me an m + t + 1. So if I take an m + t out of here, there was one m + t left. If I take the m + t here, there's a plus one. Now I've got the numerator and the denominator, both monomials. So this m + t here and this m + t here are going to cancel for a final solution of m -, t / m + t + 1. Our last example on this page is going to have to have some rearranging occur. If I pulled out common factors in these first two, I'd end up with XS left, but there aren't any XS in the end. So we're going to regroup and we're going to say let's put the cube portions together because we know a formula for the two different cubes. Then let's put the X portions together because we know a formula for the difference of two squares. The bottom is just going to factor into X -, y * X -, y. So this top X ^3 -, y cube, the formula says X minus yx squared plus XY plus y ^2, and then that's going to be plus X ^2 -, y ^2 is X plus yx minus Y, and that's all over the X -, y * X -, y. That tops not a monomial yet, but we're down to two terms when we started with four. The two terms each have an X -, y in common, so if I pull that out, I can see I get X ^2 plus XY plus y ^2 + X + y left all over the X -, y * X -, y. So one of those X -, y is in the numerator and denominator are going to cancel. I'm going to write it in descending order, so X ^2 + y ^2 plus XY plus X + y all over X -. Y is our final answer. Thank you and have a wonderful day.