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. 1/x(x-1)

Accepted Answer

Welcome back, I am so glad you're here. We are given the expression 14 divided by why times, why -7. And we are asked to express this in its partial fraction decomposition. Alright. We recall from previous lessons on partial fraction decomposition that we are going to take a look at this denominator and we are going to ask ourselves some questions about its factors. So the first question is, are these factors linear or quadratic? And both of those wise have a degree of one. So they are both linear. The next question is are these factors distinct or repeating and they're different from each other. So they are distinct. And the third question is how many factors are there? There are two of them and because there are two, we are going to set this fraction equal to two new fractions added together. And for the denominators of our two new fractions, we are going to use the factors from the denominator from our original fraction. So for the first new fraction, the denominator will be equal to the first factor or why? And for our second new fraction, the denominator will be equal to our second factor or why -7. And because both of these are linear factors, their numerator are going to be constant terms. So for the first new fraction, the numerator is going to be a capital A and for the second new fraction, it will be a capital B. And we have set this up, we are now going to go through and solve for A and B and fill them in back here and this will be its partial fraction decomposition. Alright, solving for A and B. The first step is we are going to get rid of these fractions in order to do that, we're going to multiply everything by the lowest common denominator, which is the denominator from the left hand side of the equal signs. We're going to multiply everything times Y times Y minus seven. So this will look like a times Why? Times Y minus seven all over, Y times Y minus seven. On the left hand side of the equal sign that equals a capital A times Y times Y minus seven. All over. Just a Y plus capital B. Times the Y. Times y minus seven All over Y minus seven. And now the best part when we have a fraction that has a factor in both the numerator and the denominator. Then that factor cancels out. So on the left hand side of the equal sign, the wise and the y minus seven's cancel out on the right hand side of the equal sign, the Wise cancel out for the first fraction and the y minus seven's cancel out for the second fraction. Yea, goodbye fractions. So now this will read, I'll write it more to the right hand side, 14 equals a times y minus seven plus B. Times why? And now I'll distribute that A. To the y minus seven, so 14 equals A Y minus seven A plus B. Y. Now I'd like to group these Y terms together. So this will read 14 equals A Y plus B, y minus seven. A. And now I'd like to factor a Y. Out of our Y terms. So we have 14 equals Y times A plus b minus seven. A great. So now we recall from previous lessons on partial fraction decomposition that the coefficient 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. So A plus B is going to be equal to the coefficient of Y on the left hand side of the equation but there is no Y on the left hand side of the equation. So the coefficient of that Y. On the left hand side is zero, so A plus B equals zero. Likewise the constant terms on either side of the equation are equal to each other, so the constant term on the right hand side is negative seven A. And that is equal to the constant term on the left hand side of the equation which is 14 and now we have two equations giving us a system of equations that can help us solve for A and B. Let's solve for a first we have the second equation negative seven A equals 14. Let's divide both sides by negative seven and so A equals 14 divided by -7. That's a negative two solved for A. Now we can fill A in to our first equation to solve for B. So instead of A plus B it's negative two. Plus B equals zero. Well we can add two to both sides and then we get B equals zero. Plus two is two. And we have solved for A. And for B. And now we can fill those in back up here for our partial fraction decomposition. So instead of A over Y it is now going to be negative two divided by Y. And instead of B over y minus seven is going to be two divided by y minus seven. We look at our answer choices and although these two have flipped in order, this is going to match with our answer choice B. Putting that two over y minus seven first and then minus the two divided by Y. Well done. We'll catch you on the next one.

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 1/x(x-1)

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