 This video is going to talk about multiplying and dividing rational functions. So let's remind ourselves how we multiply just regular fractions, because it's going to be the same method, just a little more complicated again. So when we multiply, there's a couple ways we can multiply. We know that the rule says you multiply straight across the top and straight across the bottom. So 3 times 10 would be 30, and 5 times 9 would be 45, and then you always have to reduce your fractions, and when you reduce it, 30 and 45 both have 15 in common. So 15 times 2 over 15 times 3, which will then give us the reduced fraction of 2 thirds. Another way that you could have done this problem was back here at the beginning, if you have common factors, numerator and denominator, because we're multiplying, we could just start reducing right away. So I see this 3, and I say, oh, down there at the bottom there's a 9, which is actually 3 times 3. So 1, 3 will cancel one of those 3s, and 5, well, 10 is actually 5 times 2. So the 5 will cancel the 5, and I'm left with 2 over 3, and I'm reduced as I multiply. So we've removed the vibrational functions. Okay, we've had to factor in the numerators and denominators separately if they can be factored. And then we're going to cancel any of the common factors, numerator and denominator, and then when you're done, you multiply the numerators across, and you multiply the denominators across, and you can leave it in factored form. And you just want to make sure, you might have reduced it, but you might have missed something, so double check that everything is simplified when you get done. So we're looking here, and I look at 3, and none of the numbers down on the bottom will reduce with 3. And 21, I'm looking at this one, and 14 and 21 both have 7 in them. So this is actually 7 times 2, and this is 7 times 3. So the 7s cancel. I'm now ready to look at my x to the 5th, and there are two x's, two factors of x on the bottom. So two of those will cancel, take off two of those, leaving me with x cubed. So now I'm just ready to multiply, so I just list everything I have left. So 3 times 3 times x cubed, I just want to make sure I got them all. And on the bottom I have this 2 times the 4, so that really gives me 9x cubed over 8. Ooh, this looks scary. But remember, we're going to factor, and that'll make it look a little less scary. So start factoring. So x squared minus 25 is x plus 5x minus 5, and we factored this x squared minus x minus 12 many times, that hopefully by now you know that's x minus 4 and x plus 3, and over to the other fraction. I have x and x, and factors of 16 that will add up to negative 8 must be negative 4 and negative 4. And then on the bottom I have a common factor of 4, leaving me with x plus 5. I usually work just straight across the top. So x plus 5, oh, I see an x plus 5 on the bottom, so I can cancel those. x minus 5, none on the bottom. x minus 4, yes, I have one x minus 4 on the bottom. And do I have another x minus 4 down there? Nope. But I've gone all the way across both numerators, so I'm ready to write down what I have left. x minus 5, x minus 4, and on the bottom we have x plus 3, and we have a 4, which we usually put in front, so I'll put it in front, and that would be our final answer. So how do you divide? Again, we're going to just start with regular fractions. So if you remember, when you divide, you take the reciprocal of the second fraction, and once you take the reciprocal, it's like the opposite, then we can do the opposite operation. So we have 3 over 5 and times now 7 over 12. We're doing the inverses here. So we invert our fraction, and that allows us to do the inverse operation. And I'm looking at this and I'm saying, oh, I can reduce, because 3 goes into 12, leaving me with 4. So on the top, I just have just the 7, it would be 1 times 7, and on the bottom, 5 times 4 is going to give me 20. So once you flip the fraction, it's everything we've been doing to this point. So when we divide rational functions, multiply by the reciprocal of the second fraction, flip it, and I say do this first, flip it first, then you won't forget to flip it. A lot of people like to start factoring and then they forget to flip it, and they think to multiply and then bad things happen. So I am quickly just going to rewrite this one as x cubed over 5x minus 15. You don't have to rewrite the first one if you don't want to, but remember that these are the two that we're going to look at now. We're not going to work with this one anymore. So let's begin factoring. 6 is a common factor of that first fraction numerator, and I have x minus 3. And on the bottom, I just have plain old 5x, can't factor that one. Times an x cubed can't be factored. And on the bottom, we have a common factor of 5 and an x minus 3. So 6. I'm going through now to see if I have any common factors. 6, no common factors in the bottom. x minus 3, yep, I have one on the top and one on the bottom. And then this x cubed, I said it couldn't be factored, but it actually can be factored to x times x times x, but we're not going to write all those. But I do see that I have one factor of x down here, so I get to take off one factor that's going to leave me with x squared. And now I'm going to clear across my numerator, so I'm ready to start multiplying. 6 times x squared on the top, and 5 times 5 on the bottom will give me 6x squared over 25. So now we have the ugly looking one. But just remember that the first thing we're going to do is write this as a multiplication, and we have x cubed plus 6x squared, and then over x squared, that used to be on the top, plus 13x plus 42. All right, so now let's start factoring. The first one will factor x plus 7 and x minus 7, and on the bottom we have a common factor of 7, and x minus 7 is the other factor. And this one has a common factor of x squared, leaving me with x plus 6. Take the smallest exponent, remember. And then here it's an x squared first term, so I know that I need factors of 42 that are going to add up to 13. And the first factors I think of are 6 and 7, and those are exactly what work. And 6 plus 7 is 13. So now we're ready to reduce. x plus 7 on the top and 1 on the bottom, so we can reduce those. x minus 7, there's 1 on the top and 1 on the bottom, so we can reduce those. x squared, no x is on the bottom. x plus 6, yes I have 1 on the top and 1 on the bottom. So the only thing I'm left with then is x squared over 7.