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. (3x+16)/(x + 1) (x − 2)²

Accepted Answer

Welcome back. I am so glad you're here. We are given the expression eight Y plus nine divided by Why plus three times y minus five squared. And we are asked to write this expression in its partial fraction decomposition form. And we don't have to solve for the constants. Yay, all right. So recalling previous lessons on partial fraction decomposition, we are going to look at our expressions denominator and it is already factored. Great. So we are going to ask ourselves two questions about those factors. The first question is are they linear or quadratic And why? Plus three that is linear that Y has a degree of one And Why? -5? That Y also has a degree of one. So that is also a linear factor. Great Question two About our denominators factors is are these distinct or repeat it? And so why plus 3? There's only one of those. So that one is distinct, But that why -5? That one is squared, that one is repeated. So we've got distinct and repeated factors here. So now that we've answered those two questions, we set this equal to and we ask ourselves how many factors are there? There are three. That why plus three and then that why -5 that repeats. So we're going to have three fractions and we'll work on the denominators first. The denominator for our first fraction is going to be our first factor. Why plus three? That one is distinct. So that one is all we're dealing with the Y plus three factor that why minus five, That one is repeated. So our second fraction over here on the right hand side. The first denominator is going to be the factor one time. Why minus five not repeated. And then our final fraction over here is going to be that Y - squared this time it will be repeated. All right, those are our denominators. Now we deal with that first question linear or quadratic. And because all three of our factors are linear, then our numerator are all going to be constant terms. So the numerator for Y plus three is going to be a. The numerator for y minus five is going to be B. And the numerator for y minus five squared is going to be C. And because we don't have to solve for the constants for A B and C. We're done. 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. (3x+16)/(x + 1) (x − 2)²

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