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. (x^2−14x+49)/(x^2−49)

Accepted Answer

Welcome back. I am so glad you're here. We are given this rational expression. The numerator is X squared minus eight. X plus 15 and the denominator is x squared minus five. X plus six. Were asked to do two things first. We need to figure out what numbers would make this expression undefined and then we need to simplify well figuring out what would make this expression undefined. We have a fraction and whenever we have a fraction, if our denominator is equal to zero then the expression is undefined. So in order to find out what numbers for X would make this undefined, we're going to set the denominator X squared minus five. X plus six equal to zero will solve for X. And we'll find out what values would make this true would make it zero in the bottom and therefore undefined. Alright so let's figure this out. By factoring what times what would give us X squared? That would be an X. Times and X. And what two numbers when multiplied, would give us a positive six. But when added together would give us a negative five. That would be a minus two and a minus three. Now we can set both of these factors equal to zero to solve for R X. And at the first one X minus two equals zero will add two to both sides. And for the second one, X minus three equals zero will add three to both sides. So our first one will be X equals two and the second one will be X equals three. And what does that tell us? That tells us that if x equals two or x equals three will have a zero in the denominator and the expression will be undefined. So X can't equal to an X can't equal three. We've solved for the first part. Now we need to simplify. Okay, going back to our expression, the numerator was x squared minus eight X plus 15. The denominator. We can write it as those factors that we solved for X minus two times x minus three. Because if we have a factor that's in common in both the numerator and the denominator, those can cancel out. And then we simplify. So let's also factor out the numerator. We have x squared minus eight X plus 15. We'll see if there has any factors in common with the denominator. So what times what gives us X squared. That's X times X. And what? Two numbers when multiplied, give us a positive 15 but added together give us a negative eight. That's going to be a minus three and a minus five. So now our expression reads in the numerator replacing it with factors we have x minus three times X minus five over x minus two times x minus three. Do we have any factors that occur in both the numerator and the denominator and therefore can cancel out. Yes, these x minus threes do. So now this simplifies to x minus five in the numerator divided by x minus two and there's our simplified expression and that's why it's important to solve for the numbers that make it undefined first. Looking at this simplified expression, yeah, we can see that X can equal to but it's not obvious at all that X can equal three. So that's why we had to do that. Step first. Alright. We look at our answer choices and this matches with answer choice. D. Well done. We'll catch you on the next one.

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x^2−14x+49)/(x^2−49)

Key Concepts

Rational Expressions

Factoring Polynomials

Domain Restrictions

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