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)(x-3)

Accepted Answer

Welcome back. I am so glad you're here. We are given the expression why divided by why -4 times. Why -8. And were asked to express this in its partial fraction decomposition. Alright. We recall from previous lessons on partial fraction decomposition that the first thing we do is take a look at the denominator and its factors. And we ask ourselves if these factors are linear or quadratic and they both have a degree of one, the wise only have a degree of one, so they are linear. Alright, we've got linear factors. And now we ask ourselves if they're repeating why -4? Y -8. No, those are distinct from each other. They do not repeat. So what we do is set this equal to a capital A. In the numerator over our first Distinct linear factor, which is why -4 plus B. In the numerator. And the denominator are second distinct linear factor, which is why -8. And there we go. We have set this thing up. Now all we have to do is solve for A. And B. And then we'll go back and fill them in. So two sulfur A. And B. We need to get rid of this fraction and to do that will multiply everything by the common denominator. The common denominator is the denominator on the left hand side. So that'll be why minus four times y minus eight. And now we'll distribute that. So that will look like why times why minus four times Y minus eight. All over Y minus four times y minus eight on left hand side of the equation that equals a times the y minus four Y minus eight. Still over y minus four in the denominator plus B times our y minus four times y minus eight. And that is all still over y minus eight in the denominator. Alright now we can do the best part, cancel things out, Okay, on the left hand side all of those y minus four's and y minus eight cancel out for R. A. The y minus four is cancel out and for R. B the y minus eight's cancel out. Now we can rewrite this as Y equals on the left hand side and then on the right that will be a times y minus eight Plus B. Times Y -4. And now we can go ahead and distribute the A. And the B. So bringing down Y equals and that'll be a Y A - A distributing the B plus y B minus for B. And now I want to regroup these so that we can bring our Y terms next to each other and then our constant terms next to each other. So this will be Y equals and then we'll have Y A plus y B. And now the constant terms minus eight a minus four B. And now with the y terms we can factor out of why and then we'll just group these constant terms together. So why equals again. And then it'll be A Y. Times A plus B. And then plus, I'm grouping these two constant terms together negative eight A minus four B. Okay, now we really need to recall what we've learned about partial fraction decomposition. We know that the coefficient of the Y term on the left hand side is equal to the coefficient of the Y term on the right hand side, but it's not super straight forward here on the left hand side, the coefficient of Y is an unwritten one. And on the right hand side, the coefficient of why comes after it. It's that A plus B factor. And since we know that those two things are equal to each other, we can write an equation. We can say that the coefficient of why? On the right hand side or a plus B is equal to the coefficient of Y. From the left hand side, it's equal to one. There we go. We've got one equation. Now we want to write an equation for our constant terms. So the constant terms on the right hand side are this negative eight a minus four B. But what are the equal to what are the constant terms on the left hand side? Well, there's nothing there and nothing. Well that's zero. So we know that the constant terms from the right hand side, negative eight a minus four B are equal to the constant terms from the left hand side, which is nothing zero. Well now we have got two equations. First equation A plus B equals one. Second equation negative eight A minus four B equals zero. And with that we can solve a system of equations and solve for A and B. Let's go ahead and do that. A lot of different ways you can go about doing this, but I am going to get the B terms to cancel out so that we can solve for a. So I'm going to take equation one and multiply it by four and so distributing the force. So I have a new equation 1. 4 times a is for a plus four times B is for B equals four times one is four. And now adding these two equations together to and the rewritten equation one negative eight A plus four A. Is a negative for a negative four. B plus four. B cancels out. That's nothing that zero and zero plus four equals four. Now I can solve for a going to divide both sides by -4 and we have A equals on the left hand side and on the right hand side, four divided by negative four is negative one. So we've solved for A A equals negative one. Great. Now I'm going to fill that in to our equation one that will give us a negative one Plus B equals one and solving for B will add one to both sides and that gives us B equals one plus one is two. We've got an A and we've got to be wonderful. Now we can go back up to this original partial fraction decomposition and fill in our A and R. B. So A would be a negative one Over Y -4 Plus B is two Over Y -8. Now we look at our answer choices and nothing exactly matches this, and that's just because they've switched the two terms around so that it's two over y minus eight minus one over y minus four, which matches with answer choice A Well done. We'll catch you on the next one.

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

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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