In Exercises 1–8, write the form of the partial fraction decompos...

Question

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (6x^2-14x-27)/(x+2) (x − 3)^2

Accepted Answer

welcome back. I am so glad you're here. We are given the expression eight X squared minus 39 X minus divided by x plus four times x minus seven squared. And we are asked to rewrite this expression in its partial fraction decomposition form but we don't have to solve for the constants. Yay, alright. We recall from previous lessons on partial fraction decomposition that we're going to take a look at the denominator. The denominator here is already factored for us. Great. So we're going to ask ourselves two questions about those factors. First question is are these linear or quadratic and X plus four that X has a degree of one? That is a linear factor and x minus seven. That X also has a degree of one. So that is also a linear factor. Alright, we've answered the first question. Now the second question is, are these factors distinct or repeated? X plus four. There's only one X plus four. Factor that one is distinct, X minus seven. That one is squared. There are two x minus seven's so that one is repeated. Alright, we've answered those questions so now we can move on. We set this equal to and now we ask ourselves how many factors are there? There are three, there's the X plus four, the x minus seven and another x minus seven. So we're going to set up three fractions here, we'll deal with the denominators. First, the denominator for our first fraction is going to be that first factor that X plus four, that one doesn't repeat. So we move on to the second factor The X -7 that one is repeated. So what we do for our second fraction here we put the X -7 once. Not repeated, just X -7. And then for our third fraction we will do X - squared. Those are our denominators. Now we'll deal with the numerator because all three of our denominators, all three of those factors are linear. All three of our numerator hers are going to be constant terms. So for X plus four the numerator will be A. For x minus seven, the numerator will be B. And for x minus seven squared, the numerator will be C. We look at our answer choices and this matches with answer choice B. Well done. We'll catch on the next one.

In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (6x^2-14x-27)/(x+2) (x − 3)^2

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