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. 6x^2-x+1/(x^3 + x²+x+1)

Accepted Answer

Hello. Today we're going to be performing partial fraction decomposition on the given expression. So what we are given is four X squared plus two X minus 18 over X plus three times X squared plus three. Now, before we perform our partial fraction decomposition, we always want to check to make sure that our denominator is fully factored here. X plus three is in its simplest form an X squared plus three is not a fact arable quantity. So let's just take a look at what is given to us in the denominator, X plus three is a linear quantity because it's leading coefficient has a power of one. So X plus three is gonna is gonna produce the partial fraction A over X plus three. And now let's take a look at X squared plus three, X squared plus three is a quadratic quantity because it's leading coefficient has an exponent of two. So X squared plus three is going to produce the partial fraction of B, X plus C over X squared plus three. Now we want to go ahead and sulfur are unknown coefficients A, B and C. But we first want to get rid of all of these denominators. So in order to do that, we're going to multiply everything by the L. C. D. And the L. C. D. In this case is X plus three times X squared plus three. Now let's go ahead and multiply everything given to us. We have four X squared plus two, X minus 18 over X plus three times X squared plus three. And we're multiplying this by the L. C. D. Which is X plus three times X squared plus three. And we're gonna go ahead and put that over one. And that is equal to the partial fraction. A. Over X plus three times the L. C. D. Which is X plus three times X squared plus 3/1 plus the partial fraction. B. X plus C over X squared plus three times the L. C. D. Which is X plus three times X squared plus 3/1. Now multiplying the left hand side by the L. C. D. Is going to completely cancel the denominator on the right hand side. Multiplying the first partial fraction is going to cancel out the X plus threes but is gonna leave us with the quantity X squared plus three. And on the second partial fraction multiplying this by the L. C. D. Is going to cancel the X squared plus three but leave us with the quantity X plus three. So what we have is four X squared plus two. X minus 18. Equal to a times X squared plus three plus the quantity B. X plus C. Times X-plus three. Now in order to continue this we need to go ahead and simplify the right hand side of this equation. And in order to do that we're gonna molt and we're gonna factor in a two X squared plus three and we're gonna foil bx plus C. With X plus three. Doing this is going to give us four, X squared plus two, X minus 18 is equal to a X squared plus three A plus the foil of bx plus C. And X plus three. Which is going to give us B x squared plus three, B x Plus CX- Plus three C. Now this is starting to become a little bit messy but what we want to go ahead and do is group up all of our x square terms, all of our X terms and all of our constant terms together. So we're going to have four X squared plus two X minus equal to a X squared plus B, X squared plus three, B. X plus C. X plus three A plus three C. And now we can go ahead and factor out an X squared from the first group and an X from the second group. And doing this is going to give us four X squared plus two, X minus 18 is equal to x squared times the quantity A plus B plus x times the quantity three B plus C plus are constants which is three A plus three C. Now, what we want to go ahead and do is create a system of equations from this given equation. And in order to do that, we're gonna be equating the coefficients of x squared terms together the coefficient of our X terms together and are constants together. So the first equation we end up with is the coefficient of x squared on the right hand side, which is a plus B Equal to the coefficient x square on the left hand side, which is four. Our second equation is going to be the coefficient of X on the right hand side, which is three B plus C equal to the coefficient of X on the left hand side, which is to and our third equation is going to be our constants on the right hand side, which is three A plus three, C Equal to the to the constant on the left hand side, which is -18. So what we have here is a system of equations involving three equations at this point, we should be comfortable with solving a system of equations. And if we solve the system of equations, we end up with the constance A is equal to one, B is equal to three and C is equal to negative seven. So now what we need to do is go ahead and plug this into our original partial fraction decomposition. Now recall that the original partial fraction decomposition was a over X plus three plus bx plus c over x squared plus three. So we're gonna go ahead and replace A with one. So what we have is one over X plus three, we're gonna replace B with three, which is going to give us three X. And we're going to replace C with -7 and that gives us the numerator three X minus seven. And this is going to be the complete partial fraction decomposition of our given statement. And that means the answer to this problem is going to be C. 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. 6x^2-x+1/(x^3 + x²+x+1)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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