Perform each long division and write the partial fraction deco...

Question

Perform each long division and write the partial fraction decomposition of the remainder term. (x⁵+2)/(x²-1)

Accepted Answer

Hello. Today we're going to be using long division to simplify the given expression and then using partial fraction decomposition to expand its remainder. So what we are given is why to the 5th plus three over Y squared minus nine. And now we need to go ahead and set up the long division. So to set the long division we have y squared minus nine, dividing our numerator, but notice that we don't have a Y to the fourth, power Y to the third, Y to the second or Y to the first. So we need to go ahead and include that in our long division. And doing that is going to give us Y to the fifth plus zero Y to the fourth plus zero Y cubed plus zero Y squared plus zero Y plus three. And now we can go ahead and perform the long division. So what we want to go ahead and do is multiply our quantity Y squared minus nine by a term. That's going to allow the first term to match up with the first term of the division. And we can go ahead and multiply y squared by y cubed to give us Y to the fifth. So again we're multiplying by Y cubed To give us way to the 5th plus negative nine times Y cubed which is going to give us negative nine Y cubed. So we have minus nine Y cubed and we can go ahead and put in a zero Y to the fourth since we don't have a Y to the fourth term here. Now recall that we need to go ahead and subtract this value in order to simplify this so we need to distribute the negative sign along this quantity here and doing so is just going to change the sign of each of our terms. So what we end up with is negative Y to the fifth minus zero Y to the fourth plus nine Y cubed. And now we can go ahead and simplify y to the fifth minus Y to the fifth is going to cancel each other out. We have no zero Y to the fourth term and zero Y cubed plus nine Y cubed is going to give us nine y cubed and now we're just gonna go ahead and bring down the next term which is zero y squared. And we're going to repeat this process now again we want to multiply y squared minus nine by a quantity that's gonna allow y squared to be as close as possible to nine y cubed. And we can multiply this by nine y. So nine y times Y squared minus nine is going to give us nine yq -81 Y. And we can go ahead and just add in a zero y squared since we never had one to start with. And we're going to go ahead and bring down the zero Y terms since we have a wide term on the bottom. Now again we're subtracting this quantity so we need to distribute the negative sign and doing so it's just going to change the signs of each of the terms. So what we have is negative nine, Y cubed minus zero, Y squared plus 81 Y. And simplifying this, the nine y cubed is going to cancel out. We don't have any y square terms and zero Y plus 81 Y is going to give us 81 Y and we bring down the next term which is going to be positive three. Now there's nothing that we can multiply y squared minus nine to give us as close as possible to 81 Y plus three because the exponent of our denominator is greater than the exponent of our numerator. So 81 Y plus three is going to be a remainder. So the long division is going to give us y cubed Plus nine, Y plus 81, Y plus three over Y squared minus nine. Now, since we have the remainder, we want to go ahead and perform partial fraction decomposition. Since that's what the problem asks us to do so again, the remainder is 81 Y plus three over Y squared minus nine and we want to go ahead and do partial fraction decomposition on this. Now, before we start the partial fraction decomposition, we always want to check to make sure that our denominator is fully factored here, we have y squared minus nine, which is a difference of squares. So we can go ahead. In fact to the denominator and that'll give us 81 Y plus three over Y minus three times Y plus three. And now we can go ahead and begin the partial fraction decomposition. Why -3 is a linear quantity as its leading coefficient has an exponent of one. So that's going to give us the partial fraction of a over Y -3. And Y plus three is a linear quantity as well as its leading coefficient has an exponent of one. So what that's gonna give us is B over Y plus three. Now we want to go ahead and perform the partial fraction decomposition in order to solve for A and B. But in order to do that, we need to first get rid of all these denominators. And in order to get rid of denominators we can multiply by the lowest common denominator, which in this case is going to be Y -3 times y plus three. So let's just go ahead and write that out. We have 81. Y plus three over y minus three times Y plus three, multiplied by the by the L C D. Over one. And that is equal to our first partial fraction which is a over y minus three times R. L C. D plus our second partial fraction which is B over Y plus three times our L C. D. And multiplying the left hand side by the LC. D. is going to completely cancel out the denominator on our first partial fraction. It's going to cancel out the denominator but leave us with why -3 in the numerator? Oh I'm sorry I cancel out the wrong the wrong quantity is gonna cancel out why minus three in the denominator and numerator. And leave us with eight times Y plus three. And on our second partial fraction we're going to cancel out the Y plus three in both the numerator and denominator. And leave us with B times y minus three. And so what we have is 81 Y plus three is equal to a. Times Y plus three Plus B Times Y -3. And now we want to go ahead and simplify the right hand side. And in order to do that we're just gonna go ahead and distribute A into y plus three and distribute B into y minus three. And doing so is going to give us eight times why? Plus three A plus B, y minus three B. And we're gonna go ahead and group up our constants and white terms together and doing so is going to give us a Y plus B, Y Plus three A -3 B. Now one more thing that we can go ahead and do is factor out the Y in the first group and that's going to give us 81 Y plus three is equal to Y times A plus B Plus three A -3 B. Now we can go ahead and use this equation to set up a system of equations. And in order to do that, we're going to equate the coefficients of our white terms together and equate the constants together and so doing this, we're gonna get two equations. The first one is the coefficients of Y, which on the right hand side is A plus B. And set it equal to the coefficient of Y on the left hand side, which is 81. So what we get is A plus B Is equal to 81. And for a second equation we're just gonna go ahead and set the constants equal to each other. So that's going to be three a minus three, B is equal to three. Now, at this point we should be comfortable with solving a system of equations. And once you solve the system of equations, you're going to get the values is equal to 41 and B is equal to 40. And now what we wanna go ahead and do is plug that into our original partial fraction decomposition. Now recall that the partial fraction decomposition we came up with is a over Y -3 -3 plus B Over Y-plus three. Now replacing a with 41 is going to give us 41 over Y -3 And replacing be with 40 is going to give us 40 over Y plus three. And this is going to be the complete partial fraction decomposition of the remainder but keep in mind that we also got a y cubed plus nine Y from the long division. So putting all of our terms together is going to give us y cubed plus nine, Y plus 41 Over Y -3, plus 40 Over Y-plus three. And this is going to be the complete expansion of our original quantity. And that means that the answer to this problem is going to be be so, I hope this video helps you in understanding how to perform the partial fraction decomposition. And I'll go ahead and see you all in the next video.

Perform each long division and write the partial fraction decomposition of the remainder term. (x⁵+2)/(x²-1)

Key Concepts

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