In Exercises 43–46, perform each long division and write the part...

Question

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^4-x^2+2)/(x^3-x^2)

Accepted Answer

Hello. Today we're going to be performing long division on the given expression and then expanding its remainder by using partial fraction decomposition. So the expression given to us is S to the power of four -16 s squared plus eight over S cubed minus four S squared. Now let's go ahead and set up the long division. And that's the first step to solving this type of problem setting up the long division is going to give us s cubed minus four S squared divided by As to the 4th minus 16 S squared plus eight. Now notice that in our given numerator we're missing an s cubed term. And for long division we must include every exponent of S. So we need to go ahead and include the s cube term by adding zero S. Cute. And so we're going to be dividing as to the power of four plus zero S cubed minus 16 X squared plus eight by a given denominator. So let's go ahead and start this. Now we want to go ahead and find a quantity where s cubed minus four squared is closest to the first two terms of this division. So if we multiply this quantity by S. What this gives us is as to the fourth minus for s cubed. Now keep in mind that we need to go ahead and subtract this quantity in order to subtract the quantity we're going to distribute the negative sign to our subtraction quantity and distributing the negative sign is just going to change the values or at least the sign of the values. And what this gives us minus S to the fourth plus four S cubed. This is going to cancel our as to the fourth term and leave us with four s cubed and now we bring down the next term which is negative 16 S squared and continue this process. Now we want to go ahead and multiply s cubed minus four S squared by a number that's going to get as close as possible to four s cubed minus 16 S squared. And the only thing we can multiply this by is positive four. So if we multiply four to s cubed minus four squared, that's going to give us four s cubed -16 as squared. And of course we need to go ahead and subtract this quantity so we need to distribute the negative sign to this quantity and doing this is just going to change the sign of all the terms. So we have minus four X cubed plus 16 as squared. And that's just going to go ahead and cancel out everything here. But we need to bring down the next term which is eight. Now there is no number that we can multiply our denominator by, that will give us a single term, so eight is going to be the remainder in this case. So the long division is going to give us s plus four plus eight over s cubed minus four S squared. Now keep in mind that the problem asks us to perform partial fraction decomposition on the remainder, so the remainder in this case, or at least this is the completion for step one, the remainder is eight over S cubed -4 S Squared. And this is what we're gonna be doing or using to perform partial fraction decomposition. Now, what we want to go ahead and do is make sure that our denominator is fully factored and if you notice, we have s cubed as our first term and S squared as our second term, so we can go ahead and at least factor out S squared in our denominator and that's going to rewrite the remainder as eight over S squared times S minus four. And this is going to be the fully factored version of our given denominator. Now we can go ahead and start the partial fraction decomposition. So let's go ahead and take a look at each of the terms. S squared is a repeating linear quantity, as S squared is equal to S times S and s by itself is a linear quantity. So we're going to get to partial fractions from S squared, the first one being a over S and the second one being be over S squared. And on the second quantity we have S -4 and this is a linear quantity because the leading coefficient has an exponent of one. So that's going to give us the partial fraction of C over S -4. Now, what we want to go ahead and do is solve for A. B. And C. But in order to do that we need to first get rid of all of these denominators in order to do that, we can multiply everything by the lowest common denominator. And in this case here the lowest common denominator is going to be S squared times S minus four. So writing this out is going to give us eight over S squared times S minus four times the L. C. D. Which is S squared times S minus 4/1. And that's going to be equal to our first partial fraction which is a over S times the L. C. D. Plus our second partial fraction which is B over S squared times R. L. C. D. Plus our third partial fraction which is C over S minus four times the L. C. D. Now multiplying the left hand side by the L. C. D. Is going to completely cancel out the denominator on the first partial fraction, multiplying the L. C. D. Is going to cancel out the denominator but only cancel out one of the S. Is in the numerator. So what we're left with is eight times S times s minus four. Multiplying the second partial fraction by the L. C. D. Is going to cancel the S squared in the denominator and numerator but leave us with B times s minus four in the numerator. And multiplying the third partial fraction by the L. C. D is going to cancel the denominator. Believe us with C times as squared in the numerator and what we have after that is eight is equal to eight times S, which is A. S Times S - plus B, Times S -4 plus C times S squared which is C. S squared. Now, what we want to go ahead and do is simplify the right hand side. And in order to do that, we're going to distribute a S into s minus minus four, distribute be into s minus four and CS squared is already simplified so we can just leave it as is so making our proper distributions is going to give us a s squared minus four A. S Plus B, S -4, B Plus CS two. And we want to go ahead and group up all of our s square terms, all of our s terms and all of our constants together and doing that is going to give us a S squared plus C. S squared plus -4 A.S plus B, S Plus - B. And finally we can go ahead and factor out S squared from our first group and s from our second group and that's going to give us eight is equal to S squared times A plus C plus S times negative for a plus B plus negative for B. And we can go ahead and use this equation to set up a system of equations that will allow us to solve for a B and C. Now notice that we don't have an s square term or an s term on the left hand side of this equation. So we can go ahead and add those by just putting them in with coefficients of zero, so we have zero S squared plus zero, S plus eight is equal to everything else. Now, in order to solve or set up the system of equations, we're gonna go ahead and set the coefficients of s squared on both sides of the equation equal to each other. So the first equation is going to be a plus C is equal to zero. For a second equation we're repeating the process but with our s terms, so we're going to have negative four, A plus B equal to zero. And for a third equation we're just setting the constants equal to each other. So what we'll have is negative four, B is equal to eight. And we can go ahead and use the system of equations to solve for our coefficients of a. B and C. Now we should be comfortable with solving a system of equations up until this point. So once we solve the system of equations, we should get the coefficients A is equal to negative half, B is equal to negative two and C is equal to one half and now we just need to go ahead and put this into our original partial fraction decomposition. And that's going to give us negative one half, negative one half over s minus two over S squared plus one half over s minus four. And that's gonna be the remainder of our original long division, expanded using partial fraction decomposition, but recall that we also want to include the S plus four that we got from the long division. So putting all of our terms together is going to give us S plus four plus, I'm sorry, not plus minus one half over, S minus two over S squared plus one, half Over S -4. And that is going to be the complete form of the given of the given expression. And that means that the answer to this problem is going to be C. So, I hope this video helps you in understanding how to perform this process, and I'll go ahead and see you all in the next video.

In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^4-x^2+2)/(x^3-x^2)

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