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. 4x^2+13x-9/x (x − 1)(x+3)

Accepted Answer

Welcome back, I am so glad you're here. We are given this expression two, Y squared plus five, Y minus four divided by Y times y minus two times Y plus five. And we are asked to express this in partial fraction decomposition. Alright, we recall from previous lessons on partial fraction decomposition that we're going to start off by looking at the denominator. We're going to ask ourselves several questions about the denominator. First are its factors linear or quadratic? And all three of those wise have a degree of one. So they are all linear. Next question are these factors distinct or repeating and they're all different from each other. So they are all distinct. And the last question which we've already answered, how many factors are there? There are three And because there are three, we are going to set this equal to three separate fractions all being added together. We are going to get the denominators for these three fractions by taking the factors from the left hand side of the equal sign. So the first fractions denominator is going to be the first factor. 1st 1 is why The second fractions denominator will be the second factor. Why -2. And the third fractions denominator will be the third factor. Why? Plus 5? Now, because these are all linear factors. All three of these fractions are going to have constant terms for their numerator. So the first fractions numerator is going to be a capital a. The second fractions numerator will be a capital B and the third fractions numerator will be a capital C. Alright, we have set this up now, what we're going to do is solve for a B and C and then come back up here and fill those in and this will be our partial fraction decomposition. Unfortunately, it's a little bit of a journey to solve for a B and C. But we can do it. Alright. The first step is that I want to get rid of all the fractions in order to do that, we're going to multiply everything by our lowest common denominator. The lowest common denominator is from the fraction on the left hand side. It's going to be why times y minus two times Y plus five. So now we can write that out, we've got two y squared plus five, y minus four times Why times y minus two times Y plus five All over. Why Times Y -2 times y plus five equal to a times Y, times y minus two times Y plus five. All over. Why? Plus B times y times y minus two times Y plus five All over. Why minus two plus C, times Y, times y minus two times Y plus five All over. Y plus five. Now, the most glorious parts, wherever there is a factor occurring in both the numerator and the denominator of a fraction. Those factors get to cancel out. So the first equation, excuse me? The first fraction on the left hand side of the equation. The wise, the y minus tunes in the Y plus fives cancel out first fraction on the right hand side of the equation. Just the wise second fraction, the y minus twos and the third fraction the Y plus five. So now this reads two, Y squared plus five, Y minus four equals capital A times y minus two times Y plus five plus capital B times Y times Y plus five plus capital C times Y times y minus two. And now I am going to do the distribution for these factors that are in parentheses. I'm not going to multiply them by the capital letters yet. First, I'll bring down the left hand side of this equation and then we've got equals A times. And for these first two parentheses, we're going to foil them. So we have Y times Y, which is why squared. And then for our outsides is y times five which is five, Y. And insides negative two times Y, which is minus two Y. So five y minus two, Y. Is plus three Y. And then our lasts -2 times five is -10. Then plus B times. Now distributing that Y to the Y plus five will be y squared plus five Y plus C. Times. Likewise distributing the Y to the y minus two will be y squared minus two Y. Now I'm going to distribute those capital A B and C letter. So this will read a Y squared plus three A, Y minus 10 A, distributing the B plus B, Y squared plus five B. Y. And distributing the C plus c, Y squared minus two, C. Y. Next I'm going to group together all of the y squared and then I will group together all of the wise and then I will group Together by itself. This constant term, this negative 10 a. So bringing down the left hand side, we'll get to use it soon, I promise, equals A Y squared plus B, Y squared plus C, Y squared plus three A. Y plus five B, y minus two C. Y and then minus 10. A. Next I am going to factor out the y squared from all these y squared terms and factor out the Y from all these Y terms. So the left hand side, two Y squared plus five, Y minus four equals on the right, y squared times a plus B plus c plus y, times three A Plus five B -2, C minus 10 A. Okay, we recall from previous lessons on partial fraction decomposition that the coefficients of like terms on either side of the equation are equal to each other. So this y squared on the right hand side has a coefficient of a plus B plus C. That is equal to the coefficient of why squared? On the left hand side. That's equal to two. We've got one equation set up the coefficient of Y on the right hand side of the equation, three A plus five B minus two C. That is equal to the coefficient of Y on the left hand side of the equation which is five. And the constant terms on either side of the equation are equal to each other negative 10 a. From the right hand side of the equation is equal to negative four from the left hand side of the equation. So now we've got three equations, we can use this system of equations in order to solve for A. B and C, which is the goal that we started talking about 300 years ago when we started this problem. Alright, let's solve for a first using equation three, let's divide both sides by negative 10. So we'll have a equals negative 4/10 or negative four negative 10th, which is positive 4/10 and 4/ reduces to 2/5. So we have a equals 2/5 we've got one of them writing it up here so that we can still see it when we scroll back up. Okay, now that we know what A is, we can fill that in for equations one and two and then do a little bit of simplification. So rewriting equation one instead of a plus B plus C, it'll be 2/5 C plus B plus C equals two. We can subtract 2/5. So from both sides and we've got B plus C equals and two minus 2/ is 8/5. So there's a new equation one. Let's do a little bit of work on the equation two. So this will be a three times not a but it will be three times to fifth, so plus five B minus two C equals five. Alright, three times 2/5 is 6/ plus five B minus two C equals five. And now we'll subtract 6/5 from both sides. So the left will be five B minus two C equals and the right five minus 6/5 is 19 5th. So and there's our new equation too. So I'll write that over here. Five B minus two C equals 19 5th. Because now we've got these two equations and we can use them to solve for B and C. Let's first take this first equation and we'll multiply everything times negative five and added to the second equation to get the b terms to cancel out and then we can solve for C. So negative five times B is negative five B negative five times C is negative five C equals 8. 15 times negative five. That's not too bad, That is negative eight. Now we can add these two equations together and the bees cancel out and we'll have negative seven C equals 19, 5th minus eight is negative 5th. So And now we'll divide both sides by -7. And yeah you can use a calculator for negative 5th divided by negative seven. That will give us a positive C equals positive 3/5. So alright. Now to solve for B let's go back to equation one here and we'll substitute in 3/5 for that C. Term. So we will have B plus not see but 3/5 C equals 8/ And we can subtract 3/5 from both sides. Well 8/5 minus 3/5. That equals 5/5 so B equals 5/5 or one B equals one. Yeah. We have solved for A. B and C. Now let's go back up to the top, fill those into our partial fraction decomposition and find our matching answer choice. So instead of A. It's 2/5 so it's 2/5 divided by Y. Instead of B. It is one so one divided by Y minus two and instead of C it is 3/ so 3/5 divided by Y plus five. 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 9–42, write the partial fraction decomposition of each rational expression. 4x^2+13x-9/x (x − 1)(x+3)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Polynomial Long Division

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