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. x^3 + x² /(x² + 4)^2

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression three X cubed plus two X squared divided by x squared plus nine squared. And we are asked to write 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 are going to look at our expressions denominator and this nominator is already factored. Great. Now that it's factored, we ask ourselves two questions about the denominators factors. First question is, are the factor or factors linear or quadratic and X squared plus nine that X has a degree of two. That is quadratic. We're dealing with quadratic factors. Second question we ask ourselves is are the factors distinct or repeated and X squared plus nine. There's two of those X squared plus nine squared. So these factors are repeated. Okay, we've got those two questions answered and now we can set this up in its partial fraction decomposition form. So there are two factors there, the X squared plus nine times X squared plus nine. So we're going to have to fractions. We'll focus on the denominators First, the denominator for our first fraction is going to be that first factor here, X squared plus nine and it's just going to be one of them. It won't be repeated. Now the second fraction is going to be our same factor X squared plus nine because it repeats. Now this one will be the X squared plus nine squared. So that's taking care of our repeated factors. Now we'll deal with the numerator and that's where that quadratic part comes in. Both of our denominators and our two new fractions are quadratic. So both of our numerator, hers are going to be linear for X squared plus nine. We're going to have the linear numerator A X plus B. And for X squared plus nine squared. We are going to have the linear numerator C, X plus D. And because we don't have to solve for the constants, that's it. We're done. That's our final answer. 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 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x^3 + x² /(x² + 4)^2

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