Int Algebra: Multiply-Divide Rational Expressions

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Rationals
Operations with Rationals
Multiplying and Dividing Rational Expressions Page [1 of 2]
Okay, so we have rational functions and now I want to show you how you can put them together by multiplying them together or dividing them. So let’s just jump right to an example, because really it’s just like multiplying fractions. You multiply tops and you multiply bottoms and so forth. So let’s just look at an example to really illustrate the point.
Suppose I want to multiply the following two rational expressions: 225t5t2+. And I want to multiply that by the following expression: 30 6t4+. Okay, well the trick is you just multiply tops and you multiply bottoms. So, in fact, you could write this out in the following way: I could say ()30) (6t 225t5t2++4. Now, I made a classic mistake. In fact, I made classic mistake number – it’s the subtracting mistake, it’s number 4 on our top ten list of classic mistakes. It’s a subtracting mistake. Actually, it doesn’t look like a subtracting mistake, but it’s basically a distributing thing. When you want to multiply this through, you have to actually remember to distribute. So I call it the subtracting mistake, because it’s usually to do with a negative sign there, but it’s okay, it’s the same thing. In this case, it’s 2(6t + 30). It’s not just (2 × 6t) + 30. You’ve got to remember to spread that negativity. In this case, the negativity happens to be 2. Okay, so be it. Anyway I put parentheses around there to show that I’m multiplying 2 by all that, not just that.
Okay, well now I can do some cancellation here. For example, this 2 can cancel with that 4, because they’re factors now. I’m left with a 2 here. Anything else can I do? Well, I don’t know, but I can factor some common stuff out of here and factor some common stuff out of here. So, let’s simplify this. This, it looks like I can factor out a 5t. So I factor out the common factor of 5t and I’m left with just a t here and a 5 here. You can distribute and see I have 5t2 + 25t. On the bottom it looks like I can easily factor out a 6. If I factor out a 6, I'm left with just t + 5. Oh, look at that! There’s more cancellation! We can cancel this with this and see that this just equals – look how complicated. This whole product looks so awful on the onset, but now it just equals 65t. Again, let’s make sure that we didn’t make a mistake. Ah! Okay, I made a great mistake. It’s always good to stop before you give your answer and start plugging it in as the answer, to look over and see what’s going on here.
Do you see the mistake I made? In fact, you probably saw it earlier, maybe, and that is I was writing all this stuff and I factored this – what happened to that 2? I canceled the 2 with the 4 and I had an extra 2. Whatever happened to that 2? This is the lonely 2. Now where did that go? Well, it’s not here. So I actually was supposed to write a 2 right here. This should be 2 × 5 × t. So there should be a 2 here. And now I can do even more canceling, because, in fact, the 2 and the 6 can cancel and I’m left with now just 35t, even better yet, because it’s correct. So this is the answer. So I made that mistake on purpose, just to illustrate the point that you have to be careful and make sure that everything’s been taken care of. Great!
So multiplication is not a problem, but the division is a little more important. I want to remind you how this works. If you have two things, like BA, and you want to divide it by another fraction, DC, remember what you do – you invert this and multiply. So that would be BCAD. So remember, when you’ve got this thing going on here, you invert and multiply. In fact, sometimes they're going to give it to you in a complex form, like this. But again, remember the trick is you just invert the bottom and multiply. You might be asking, by the way, “Gee, we’re always flipping everything. Should we ever flip this?” Do you ever flip the top guy? Do you ever flip the guy on the left? Let’s see. No, I guess not. There’s no flipping of that guy. That guy stays the way that it is, but you flip the bottom guy. Okay, you got all that. I finally got that joke in.
Rationals
Operations with Rationals
Multiplying and Dividing Rational Expressions Page [2 of 2]
So let’s try a quick example here, 5xx2+, and we’re going to divide that by 25 yxy +. Okay, so what do I do? Well, I invert and multiply, so this is going to equal 5xx2+ times – and now I invert, so I have yxy 25+. And now I multiply like we just discussed, which is I just put everything together like that, and so what I’d see here is x2 + x – look how I put parentheses around that, because everything has to hit the 25. And then everything has to hit the 5, so I’m left with this, 5 (xy + y). Now, I can factor a little bit here, I can factor out a common factor of an x on top. Let’s do that, x(x + 1), and I’ve got 25. And then on the bottom I can factor out a y, and so I’d have 5y(x + 1). Oh, neat! The x + 1 common factor can go away and what about the 5’s? This 5 canceled that 5 and I have 5 on top. I’m going to remember it this time, so I see y5x. And so the quotient of these two rationals turns out to be this y5x.
I think we have time for just one last one. This one’s going to be a little bit more involved. In fact, let me give you a chance to try this. This one, I have to admit though, is not a very friendly one to give to someone, but what the heck. So I want you to take 45kk23kk22++++, and I want you to divide that by 2410kk65kk22++++. Invert and multiply away, and see if you can reduce it to its lowest terms.
Okay, well, let’s see how I'm going to make out here. Actually, you know what I think I’m going to start doing, since it’s obvious that it’s going to be factoring galore here, I’m going to start to factor this fraction first, and then I’ll invert and multiply. Let’s see if we can do a lot of steps together. If you don’t like doing a lot of steps together, then just do them all separately, but I'm going to see if I can do some stuff here. I’m going to put a k here – I’m just factoring this top right now. They're both going to be the same sign, they're both going to be positive, and 2 and 1 work great, because 2 and 1 add to give 3, so that’s fine. And I divide that by the bottom – you can see the value of factoring, by the way. It simplifies things in a lot of cases. Both the same sign, they’re both positive, great. Now, let’s see, how do I combine to get 4, 2 and 2 or 4 and 1? Well, 4 and 1 is good, because if I add them together, I get 5.
Now, I’m going to invert and multiply all in one step. So now I’m going to write the bottom on top, but, at the same moment, I'm going to try to factor. See how I'm parallel processing here? Not a good idea if you really aren’t comfortable with this, by the way, but I’m okay with this at the moment. Same sign, positive. Okay, it has to multiply together to give 24 and add to give 10, so 6 and 4 sounds like a good combo to me. And now the bottom, which used to be the top and I’m flipping, because I'm dividing and now multiplying, can be factored as k and k. Same sign, positive. And what do we have here? The product is 6, sum is going to be 5, that sounds like a 2 and a 3 to me.
And now, everything is multiplied together, so I can just cancel at will here. Let’s see, is any canceling happening? Oh sure, are you kidding? After all that work? Do you think they would give us this problem if nothing would cancel? Maybe. These guys cancel away, these guys cancel right here. Oh look! These guys cancel right here, and we’re just left with what? 3k6k++ , and so that’s the answer. You can rewrite it, if you want, or you can hand in your paper just like this, including your hands.
Anyway, that’s the answer and that’s how you multiply and divide rational expressions.