Answer each question. By what expression should we multiply ea...

Question

Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?

Accepted Answer

Welcome back. I am so glad you're here. We are asked what should be multiplied by each side of. And then we're given an equation. I'll write that in a minute so that the fraction form of the equation is removed. Here's our equation. On the left hand side of the equal sign in the numerator, we have five Y minus three that's divided by, in the denominator, we have Y multiplied by the quantity of nine Y squared plus two squared. That is equal to, on the right hand side, we have three fractions. The first is a divided by Y plus the second fraction. The numerator is by plus C that's divided by, in the denominator, we have nine Y squared plus two plus the third fraction. In the numerator, we have DY plus E and that's divided by, in the denominator, the quantity of nine Y squared plus two squared. Our answer choices are answer choice, A nine Y squared plus two answer choice by multiplied by the quantity of nine Y squared plus two. Answer trace C, the quantity of nine Y squared plus two squared and answer choice Dy multiplied by the quantity of nine Y squared plus two squared. All right. So here we need to get rid of the fraction. And in order to do that, we're going to multiply everything by the factors that are contained in all of the denominators because we need them to cancel out. So the, on the right hand side, we have a Y factor. So we're going to need to multiply everything by A Y, we have a nine Y squared plus two. So we can put in parentheses multiply that Y by nine Y squared plus two. And then the third denominator is also nine Y squared plus two, but that's squared. So we need to be sure we square our nine Y squared plus two. And if we look at the left hand side of the equation, that's exactly what there is that Y multiplied by the quantity of nine Y squared plus two squared right now, you know what the answer is. But I'm going to show you how this works. So if we write this out and we're gonna keep everything in factors. So the left hand side of the equation in the numerator, we have the quantity of five Y minus three multiplied by Y multiplied by the quantity of nine Y squared plus two squared, all divided by Y multiplied by the quantity of nine Y squared plus two squared equals. Now we have our capital a multiplied by Y multiplied by nine Y squared plus two squared. That's all divided by just Y plus our second fraction. On the right hand side will have the quantity of that numerator by plus C. And that's being multiplied by Y multiplied by the quantity of nine Y squared plus two squared, all divided by nine Y squared plus two. And then we finally have our third fraction which is plus the quantity of in the numerator dy plus E which is being multiplied by Y multiplied by the quantity of nine Y squared plus two squared. That's all divided by the quantity of nine Y squared plus two squared. And now the fun part where everything cancels out. So on the left hand side, with the fraction, the Ys cancel out and the factors of nine Y squared plus two squared cancel out on the right hand side, the first fraction, the Y is cancel out the second fraction. On the right hand side, the nine Y squared plus two in the denominator cancels out one of the nine Y squared plus two factors in the numerator. And for the third fraction on the right hand side, the nine Y squared plus two squared cancels out and look there, our denominators are gone, our fractions are gone and that's how we arrive at answer. Choice. D well done. We'll catch you on the next one.

Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?

Key Concepts

Partial Fraction Decomposition

Common Denominator and Clearing Fractions

Factoring and Identifying Denominators

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