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Multiplying rational expressions

When we multiply rational expressions, we multiply both numerators and multiply both denominators. We can also see if we can reduce the product to lowest terms. This is very similar to multiplying fractions, only we also have to think about the domain while we do it. Created by Sal Khan.

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Video transcript

- [Instructor] So what I have here is an expression where I'm multiplying rational expressions. And we want to do this multiplication and then reduce to lowest terms. So if you feel so inspired, I encourage you to pause this video and see if you can have a go at that. All right, now let's work through this together. So multiplying rational expressions like this, it's very analogous to multiplying fractions. For example, if I were to multiply 6/25 times 15/9, there's a few ways you could do it. You could just multiply six times 15 in the numerator, and 25 times nine in the denominator. But the way that many of us approach it so that it's easier to reduce to lowest terms is to factor things, to realize that look, six is two times three. Nine is three times three, 15 is three times five, and 25 is five times five. And then you can realize in your eventual product you're going to have a five in the numerator, five in the denominator. You can, five divided by five is one, three divided by three is one, and then three divided by three is one. And so all you'd be left with is that two and then that five. So this is going to be equal to 2/5. So we'll do the analogous thing here with these rational expressions. We're going to factor them, all of them in the numerators and denominators. And then we'll see if we can divide the numerator and the denominator by the same thing. Now, the one thing we have to make sure of as we do that is we keep track of the domain. 'Cause these rational expressions here, they might have x values that make their denominators equal to zero. And so even if we reduce to lowest terms, and we get rid of those expressions, in order for the expressions to be the same expression, we have to constrain the domain in the same way. So let's get started. So this is going to be equal to, I'll just rewrite everything, x squared minus nine. How do we factor that? Well, that's going to be a difference of squares. We could write that as x plus three times x minus three. And then that is going to be over this business. And let's see, five squared is 25, negative five plus negative five is negative 10. So this is going to be x minus five times x minus five. If what I'm just doing here with the factoring is not making sense, I encourage you to review factoring on Khan Academy. And then we multiply that times, let's see, in this numerator here, I can factor out of four. So that's going to be four times x minus five. Which is going to be useful. I have an x minus five there, x minus five there. And then that's going to be over. Let's see, this expression over here. Two plus three is five, two times three to six. So it's going to be x plus two times x plus three. Now before I start reducing to lowest terms, let's think about the domain here. And the domain is going to be constrained by things that make these denominators equal to zero. So the domain would be all real numbers except x cannot equal five. Let me write it over here. X cannot equal five because if that happened, then this denominator would become zero. X cannot be equal to negative two. X could not be equal to negative two because that would make the denominator here zero, which would make the denominator here zero. And x cannot be equal to negative three. So the domain is constrained in this way. We have to carry this throughout. No matter what we do to the expression, this is the constraints on our domain. With that out of the way, now we can reduce to lowest terms. So x, we have an x plus three in the numerator, x plus three in the denominator. X minus five in the numerator. X minus five in the denominator. And I think we've gone about as far as we can. And so when we multiply the numerators, we are going to get, this business is going to be four times x minus three. Four times x minus three, over, we have an x minus five here. X minus five. And then we have an x plus two. We have an x plus two. And we could leave it like this if you want. In some cases, people like to multiply the things out, but we're done. We've just finished multiplying these rational expressions. And we have to remind ourselves that x cannot be equal to any of these things. Now, the way that we've simplified it, we still have an x minus five right over here. So it might be redundant to say that x cannot be equal to five because that's still the case in our reduced terms expression here. And that's true also of the x cannot be equal to negative two. We still have an x plus two here. So still, even in this expression it's pretty clear, that x cannot be equal to negative two, but the x cannot equal negative three isn't so obvious when you just look at this expression. But in order for this expression to be completely equivalent to the original, it has to have the same domain. And so you might want to explicitly say that x cannot be equal to negative three here. You could also say the other two, but those are, that's still very clear when you look at this expression.