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+4/x² (x²+4)

Accepted Answer

Hello. Today we're going to be performing partial fraction decomposition on the given expression. So the expression given to us is nine Y plus one over Y squared times the quantity Y squared plus nine. Now, before we perform our partial fraction decomposition, we always want to check to make sure that our denominator is fully factored. Why squared is in its simplest form and why squared plus nine cannot be factored? So the denominator is in its simplest form. Now let's take a look at what type of partial fractions this is going to produce. Now it's easy to confuse Y squared as a quadratic quantity. But keep in mind that why squared is the same thing as Y times Y. And why by itself is a linear quantity. So why squared is really a repeating linear quantity and that's going to produce two partial fractions is going to produce A over Y and plus B over Y squared. Now let's take a look at y squared plus nine. Y squared plus nine is a quadratic quantity because it's leading coefficient has an exponent of two. So this is going to produce the partial fraction C, Y plus D over Y squared plus nine. And now what we want to go ahead and do is sell for our coefficients, A, B, C and D. But in order to do that, we need to first find a way to get rid of all of these denominators while in order to do that we can multiply everything by the L C D. And the L C D. In this case is y squared times y squared plus nine. So let's just go ahead and write, write this out. So we're gonna multiply everything by the L. C. D. So we have nine Y plus one over Y squared times Y squared plus nine. And multiplying this by the L C. D. Is going to be Y squared times Y squared plus 9/1. And this is going to be equal to the partial fraction over Y times the L. C. D. Which is why squared times Y squared plus 9/ plus the partial fraction be wise be over Y squared times Y squared times Y squared plus 9/1 plus the last partial fraction which is C. Y plus D over Y squared plus nine times Y squared times Y squared plus 9/1. Now multiplying the left hand quantity is going to completely get rid of the denominator. And on our first partial fraction multiplying this by the L. C. D. Is going to cancel out one instance of why in the denominator and the numerator. So in the numerator we have eight times Y times Y squared plus nine. On our second partial fraction, multiplying this by the L. C. D. Is going to cancel out the Y squared in both the numerator and denominator but we still have this Y squared plus nine in the numerator. And on our third partial fraction multiplying this by the L. C. D. Is going to cancel out the Y squared plus nine. But we are left with the Y squared quantity in the numerator. So simplifying this is going to give us nine Y plus one is equal to eight times Y. Which is a Y times the quantity Y squared plus nine plus B times the quantity Y squared plus nine plus Y squared times the quantity C. Y plus D. And now in order to sulfur are coefficients. We want to go ahead and simplify this right hand side. We're going to distribute A. Y into the quantity Y squared plus nine. We're going to distribute B into the quantity Y squared plus nine. And we're going to distribute Y squared into the quantity C. Y plus D. And this is going to give us nine, Y plus one is equal to a Y cubed Plus nine, AY plus B, Y squared plus nine B plus c. Y cubed plus dy squared. Now this is looking like a jumble of terms. But what we want to go ahead and do now is group up our terms together. So we want to group up all of our white cube terms together, all of our white square terms together. All of our white terms and all of our constants together. And doing this is going to give us nine Y plus one is equal to a Y cubed plus C. Y cubed plus B, Y squared plus dy squared plus nine A. Y plus nine B. Okay. And now what we can go ahead and do is factor out the y cube term in our first group and a factor out a Y squared term in our second group. And this gives us nine, Y plus one is equal to y cubed times A plus C plus y squared times B plus D Plus nine A. Y plus nine B. And now we can go ahead and use this equation to set up a system of equations. That's going to allow us to suffer A B, C and D. And in order to do that, we want to go ahead and set the coefficients of our white cube terms are y squared terms are white terms and are constants together. But if you notice we're missing a white cube term and a y squared term on the left hand side of this equation, while we can go ahead and add them in by just giving them a coefficient of zero. So what we have is zero, Y cubed plus zero, Y squared plus nine Y plus one. And now we can create our system of equations. So our first equation is going to be the coefficients of our white cube terms equated to each other and this is going to give us a plus C is equal to zero for our second equation. We're going to equate the y squared coefficients together and this is going to give us B plus D Is equal to zero. Now we want to go ahead and equate the coefficients of our white terms together and here we have nine A is equal to nine. So our third equation is going to be nine A is equal to nine. And for our last equation we're going to set the co the constants equal to each other. So we have nine. B as a constant and one as a constant. So we have nine. B is equal to one. And now we want to go ahead and solve the system of equations to get our coefficients. But by now we should be comfortable with solving a system of equations. So if we solve the system of equations, we're going to get the coefficients of A is equal to one. B is equal to 1/9, C is equal to negative one and D is equal to negative 1/9. And now we want to go ahead and replace the A, B, C and D coefficients in our partial fraction decomposition. Now recall that the original partial fractions were A over Y plus B over y squared plus C, Y plus d over y squared plus nine. And now we want to go ahead and replace it. So we know that A is equal to one. So we're gonna go ahead and replace the first partial fraction with one and we get one over Y. We know that B is 1/9. So replacing B is going to give us 1/9 over y squared. We're gonna replace C. With negative one. So C. Is going to give us negative one why or negative why? And we're gonna replace D with negative 1/9. So that's gonna be negative y minus 1/9. Now, on the last partial fraction we can go ahead and factor out a negative and factory now a negative is going to give us the quantity why? Negative Y plus one plus 1/9, Y plus 1/9. And we can go ahead and move this negative to the front of the partial fraction. And doing this is going to give us the partial fraction decomposition of one over Y plus 1/9, Y squared minus Y plus 1/9. All over Y squared plus nine. And this is going to be the complete partial fraction decomposition of our given statement. And that means that the answer to this problem is going to be a So I hope this video helps you in understanding how to perform partial fraction decomposition. And I'll go ahead and see you all in the next video

In Exercises 9–42, write the partial fraction decomposition of each rational expression. x+4/x² (x²+4)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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