In Exercises 9–42, write the partial fraction decomposition of ea...

Question

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x^2 +2x -3)

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression eight Y divided by Why squared plus four Y minus 12. And we are asked to write this in partial fraction decomposition. Alright. We recall from previous lessons on partial fraction decomposition that we're going to take a look at the denominator and we notice with this denominator that it is not yet factored. So first step is going to be to factor this denominator. So what times what gives us Y squared? Well that's going to be why times why? In each of the first spots in our parentheses. And now we need two numbers that when multiplied give us a negative 12 but added together give us a positive four, That would be plus -2. So we can rewrite this expression as eight Y divided by Why plus six times y minus two. And now we ask ourselves three questions about the denominator are these factors linear or quadratic? And both wise have a degree of one. So they're both linear. Next are these factors repeating or are they distinct? And they're different from each other? So they are distinct. And the last question which we've already answered is how many factors are there? There are two. So we are going to set this fraction equal to two separate fractions added together. And the denominators for each of these two fractions are going to be the two factors that we have on the left hand side of the equal sign. So the first fractions denominator will be the first factor. Why? Plus six and the second fractions. Denominator will be the second factor y minus two. And because these are both linear factors than both of our numerator are going to be constant terms. So the numerator for the first fraction is going to be a capital A. And the numerator for the second fraction will be a capital B. Alright, we have set up our partial fraction decomposition. Now we are just going to solve for A and B and then fill them in back up here. It's a little bit of work to solve for A. And B. But we can do it to solve for A and B. First we're going to get rid of all of these fractions. So we're going to multiply all three of these fractions times the lowest common denominator, which is from the left hand side of the equal sign, it's the Y plus six times y minus two. So this will look like we'll have eight Y times Y plus six times Y minus two. All over, Y plus six times y minus two. Equal to capital A multiplied by Y plus six times y minus two. All over Y plus six plus capital B times. Why? Plus six times Y minus two. And that is all over. Why minus two. And now the best part of this, any fraction that has a factor occurring in both the numerator and the denominator gets to have that factor canceled out. So on the left hand side of the equal sign, the y plus sixes cancel out. And the y minus two, the first fraction on the right hand side of the equal sign, The Y plus six is cancel out. And the second fraction the y minus twos cancel out. So now this will read eight, y equals capital a times y -2 plus capital B, times Y plus six. And next I'm going to distribute the A. And the B to those factors in the parentheses. So bringing down eight Y equals a Y minus to a plus, distributing the B, B, Y plus six B. And now I want to regroup these so that ry terms are together and our constant terms are together. So this will be eight, Y equals A Y plus B, y minus to a plus six B. And the next step is to factor out a Y. For our y terms. So this will read eight, Y equals Y times A plus B. And then our constant terms -2, a plus six b. Okay, we recall from previous lessons on partial fraction decomposition that the coefficients of like terms on either side of the equal sign are going to be equal to each other. So the coefficient of why on the right hand side of the equation is a plus B, and a plus B is equal to the coefficient of y. On the left hand side of the equation, which is eight. So a plus B equals eight. Same with our constant terms. The constant terms on the right hand side of the equation negative two. A plus six B are equal to the constant terms on the left hand side of the equation. But what is the constant term? Well, there's nothing there and nothing that's zero. So negative two, A plus six B is equal to zero. Now we have two equations and we can use this system of equations to solve for A. And B. And then go back up to the beginning and fill those in. Like we talked about way back when we started solving this problem. So let's multiply everything in the first equation times two and then add that to the second equation to cancel out our A terms. So two times A is two, A plus two times B is two. B equals two times eight, which is 16. Now adding these equations together are A is cancel out and we'll have eight. B equals 16, solving for B will divide both sides by eight and we get B equals I'll write it up here so we can see it when we scroll up, B equals 16 divided by eight will be equals to great. We have solved for B. Now we can plug that back into our first equation for B. And so it won't be A plus B. It will be A plus two Equals eight and we can subtract two from both sides of the equation and we'll get A equals eight minus two. Is six. Write that up here so we can see it when we scroll up. We have solved for A and B. Now we can go back and fill those in to our partial fraction decomposition, so filling them in up here. A is being replaced with six, so we've got six divided by Y plus six, and B is replaced with two. So six over Y plus six plus two over y minus two. We look at our answer choices and this matches with answer choice. A. Well done. We'll catch on the next one.

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x/(x^2 +2x -3)

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