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 am so glad you're here. We are given this rational expression. The numerator is x minus one divided by the denominator which is X squared minus one. We're asked to do two things. We want to write all the numbers that make this expression undefined and then were asked to simplify. So we have a fraction. What would make a fraction undefined? Well it would be undefined if its denominator was zero and because we have an X in the denominator we have to ask ourselves what values for X would make our denominator equal to zero and therefore undefined. And in order to do that we can set our denominator X squared minus one equal to zero and solve for X. Several ways to do this. Let's factor x squared minus one. We look at that and we recognize we have a difference of squares. So recalling previous lessons on factoring differences of squares, we know that we are go going to take the square root of both of those two numbers. So the square root of x squared is X. So X goes in the first position in each set of parentheses and the square root of one is one. So that goes in the second position in each set of parentheses. And by definition one is plus and one is minus now we have two factors and we can set each of those equal to zero to solve for X and for the first one, X plus one equals zero. We can subtract one from both sides for X minus one equals zero. We can add one to both sides. And so simplifying we have X equals negative one and X equals a positive one. And what does this tell us? This tells us that if X does equal negative one or positive one then our denominator is going to be equal to zero and therefore undefined. So X cannot equal negative one and cannot equal positive one. Alright, we've figured out the first part of that. Now let's simplify our expression. Our numerator is x minus one. And what we want to do to simplify is see if we have any factors that occur in both the numerator and the denominator and can therefore cancel out. Well, we've already factored the denominator that's X plus one times X minus one. And to any of those factors occur in both X minus one. Does those simplify now? Those are ones. So this simplifies to one divided by X plus one. All right. Now, you can see looking at the denominator here, why it's clear that X cannot equal a negative one but it's not clear that X cannot equal positive one. And that's why we had to find the numbers that make this expression undefined first and then simplify. 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. (3x−9)/(x^2−6x+9)

Key Concepts

Rational Expressions

Factoring Polynomials

Domain of a Rational Expression

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