In Exercises 15–32, multiply or divide as indicated. (x^2+x)/(x^2...

Question

In Exercises 15–32, multiply or divide as indicated. (x^2+x)/(x^2−4) ÷ (x^2−1)/(x^2+5x+6)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this great expression here of division. We've got X squared divided by x squared minus five X plus six. All divided by X divided by two things here, X minus two times X squared minus six X plus nine. And we are to do the division and simplify. Well, first of all we recall that when we are dividing fractions we are going to flip the one after the division sign and then change that to multiplication. So let's rewrite the first one. We've got X squared divided by X squared minus five X plus six. And now in the new numerator we've got x minus two times X squared minus six. X plus nine divided by X. So that is our new rewritten one. And now when we are multiplying we want to get everything as factored as we possibly can. So we take a look at this X squared minus five X plus six. And we say hey can we factor that? Let's give it a try. So what times what gives us X squared? Will that be X times X. And what times what gives us six but adds up to be a negative five. Will that be x minus three and x minus two? Alright, and now on this second one we've got x minus two is factored but we can factor this X squared minus six. X plus nine. A little bit further. What times what is X squared? That's x times X. And we want to multiply to get positive nine but add up to be negative six. Well that would be a minus three minus three and that's all still over X. And now is the super fun part because we get to cancel things out. So looking at our denominator in the first one, X minus three down there. Is there an X minus three on the top? Yes there is. They are canceled. Those are both now just one And X -2. In the denominator over here. Is there an X -2 in the numerator? Yes. Those cancel each other out and they're just one. And what else do we have? Well, we have this X in the denominator over here, we have an X squared up here. X squared is the same as X times X. So now it's just going to be one X left in the numerator. So we've got X times X minus three because all those other things are ones. And in the denominator it's just one times one times one. Well that's one. So we don't have to worry about the denominator. We've got X times X minus three and that doesn't match our answer choices. So we can go ahead and distribute that X. We have X squared minus three X. And that does match our answer choice. D. Well done. We'll catch you on the next one

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