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. 5x²-6x+7 /(x − 1) (x² + 1)

Accepted Answer

Welcome back. I'm so glad you're here. We are given this expression seven X squared minus nine X plus 13 divided by x minus four times X squared plus four. And we are asked to rewrite this in its partial fraction decomposition form but we don't have to solve for the constants. Yay, alright. We were called from previous lessons on partial fraction decomposition that we're going to take a look at the denominator and the denominator is already factored for us. Great. So now we ask ourselves two questions about those factors. The first question is, are they linear or quadratic And X -4? That X has a degree of one. So that is linear an X squared plus four that X has a degree of two. That one is quadratic. So we're dealing with both here. The second question we ask ourselves is if our factors in our denominator are distinct or repeated, so x minus four and x squared plus four, those are not repeated. They both have occurred just once and they're not the same as each other. So they're distinct. Alright, now that we've answered those questions, we set this expression equal to and because we have two factors, it's going to be two fractions added together and we'll deal with the denominators first. The first fraction is going to be our first factor there, that x minus four and the second fraction is going to be our second factor X squared plus four. They are distinct. None of them repeat. So it's just those two fractions. Now we'll deal with the numerator Z. And that first question of linear or quadratic. Our first fraction here has X -4 in the denominator that's linear. So its numerator is just going to be a constant term. A. Our second fraction has x squared plus four in the denominator, which is quadratic because the denominators quadratic, our numerator is going to be linear, which will be B. X plus C. And because we don't have to solve for the constants for A. B and C, we're done. That's it. We look at our answer choices and this matches with answer choice C. 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. 5x²-6x+7 /(x − 1) (x² + 1)

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