In Exercises 7–14, simplify each rational expression. Find all nu...

Question

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (3x−9)/(x^2−6x+9)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression, I'll rewrite it. We have x minus one in the numerator and we have X squared minus one in the denominator. And we want to do a couple things here. We want to find all the numbers that make this expression undefined. And we're going to simplify, we're actually going to start off by simplifying pause, find out what makes this undefined and then go back to simplification. So the first step of simplification is to factor anything we can. So in the numerator x minus one, that's about as factored as it gets. But the denominator, X squared minus one. We recall from previous lessons that that is a difference of squares. So this first square X squared what times what gives us X squared. That's X times X. This second square one. What times what gives us 11 times one. And we recall that for a difference of squares, one of them is going to be a plus and one is going to be a minus. Great. We have got that all simplified into factors. And the next step of simplification would be to start canceling out any factors that exist in both the numerator and the denominator. But we have to pause. We have to screech to a halt and say before we start canceling anything out what numbers would make this expression undefined because we don't want to inadvertently get rid of a number that would make it undefined. So pausing what is undefined undefined means that there's a zero in the denominator and you can't do that, you can't divide anything by zero. So what would make a zero in the denominator. Well, in order to find that we're actually going to take each factor and set it equal to zero. It's like we have to do the opposite to figure this out. So X plus one equals zero. Will subtract one from both sides. That's an X equals negative one. X minus one equals zero. That add one to both sides. That gives us X equals one. And what does this tell us? That tells us that X cannot equal negative one or one because that will give us a zero in the denominator. Great, good sneaky work there guys, we figured out what gives us zeros, what makes it undefined. And now we can get back to our lovely simplification where again we look for factors that occur in both the numerator and the denominator and we've got one. We've got X minus one. And so those cancel out and they become ones. And now in the numerator it's just one and then the denominator it's X plus one times one which is just X plus one. And that's it. That's as simplified as it's going to get. So we look at our answer choices and that matches with answer choice. D Well done. We'll catch you on the next one

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