Find the domain of each rational expression. See Example 1.

12
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Question

Find the domain of each rational expression. See Example 1.

12
——————
 x² + 5x + 6

Accepted Answer

Hello, everyone. We are asked for the following rational expression to determine its domain. We are given 14 divided by, in parentheses. A squared plus nine, a plus 14. Our answer choices are a, the interval from negative infinity to negative seven union with the interval from negative seven to negative two union with the interval negative to 14 union with the interval 14 to positive infinity. Not including any of our endpoints B the interval from negative infinity to negative seven union with the interval from negative seven to negative two union with the interval from negative two to positive infinity. Not including any of the end points C the interval from negative infinity to two union with the interval from 2 to 7 union with the interval from 7 to 14 union with the interval from 14 to positive infinity. Not including the end points. D the interval from negative infinity to two union with the interval from 2 to 7 union with the interval from seven to positive infinity, not including any of the end points. So I'm going to rewrite this expression as a fraction. So my numerator is 14 and A squared plus nine A plus 14 is my denominator. I recall that my domain is all the X values except what makes the fraction undefined. So we don't want anything undefined and undefined fractions happen when the denominator equals zero. So to find where this would be undefined, we need to set our denominator equal to zero. So we have a squared plus nine A plus equals zero. Unfortunately, we can't just solve this, this way. We need to do one more step. We need to factor this trinomial into two binomials. So factoring A squared plus nine A plus 14, I know the first term in each of my new binomials will be a because it needs to multiply back to A squared. And then my second terms, I need two numbers that multiply to 14 and add to nine and those numbers would be two and seven. So factored, I have the parentheses. A plus two multiplied by the parentheses. A plus seven equals zero. Now I can set each factor equal to zero. So a plus two equals zero. A plus seven equals zero and solve for a to find out what numbers can't be in our domain. So A plus two equals zero. A cannot equal negative two. And for a plus seven equals zero, we find that A cannot equal negative seven. So it'll be undefined when A equals negative two or negative seven. So those can't be in our domain. But every other real number can So we're gonna start at negative infinity, open parentheses, negative infinity. And this will go to our smallest number. So negative seven closed parentheses. And then we're gonna union that with the parentheses from negative seven to negative two. And then union that with negative two to positive infinity. And that'll cover all of the real numbers except when A equals negative two or negative seven. And this matches answer choice B have a nice day.

Find the domain of each rational expression. See Example 1. 12 —————— x² + 5x + 6

Key Concepts

Rational Expressions

Finding the Domain

Factoring Polynomials

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