Find the partial fraction decomposition for each rational express...

Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-3)/(x^2(x^2 + 5))

Accepted Answer

Hello, today we're gonna be simplifying the rational expression by using partial fraction decomposition. So what we are given is negative nine over T squared times the quantity T squared plus seven. Now, in order to use partial fraction decomposition, we always want our denominator to be in a fully factored form. Currently, our denominator is given to us in a factored form. So we can immediately set up our partial fractions. The first partial fraction comes from the term of T squared T squared is a non repeating quadratic equation. And that is going to give us A X, I'm sorry, A T plus B over T squared. And for the second partial fraction, we have T squared plus seven, which is a non repeating quadratic quantity. And that is going to give us the second partial fraction of C T plus D over T squared plus seven. So now that we have our partial fraction set up, we wanna go ahead and sol for the quantities of ABC and D. And in order to do that, we're going to need to first eliminate the denominators from this given equation. So in order to eliminate the denominators, we can multiply everything by a lowest common denominator. In this case here, the lowest common denominator is going to be T squared times T squared plus seven. We're gonna go ahead and multiply each term by the lowest common denominator doing. This is going to give us negative nine over T squared times T squared plus seven, multiplied by T squared times T squared plus seven over one. And that is going to equal to A T plus B over T squared times the L CD, which is T squared times T squared plus 7/1. And that's gonna be plus C T plus D over T squared plus seven multiplied by the L CD which is T squared times T squared plus 7/1. Now, what we can go ahead and do is use cross reduction to eliminate the denominators. So in the first term, we have T squared times T squared plus seven in both the numerator and denominator. So we can go ahead and eliminate those quantities. In the second term, we have T squared in both the numerator and denominator. So we can go ahead and eliminate that quantity. And in the last term, we have T scored plus seven in both numerator and denominator. So those are going to get eliminated. This is going to leave us with negative nine is equal to the quantity A T plus B multiplied by T squared plus seven plus C T plus D multiplied by T squared. So now what we're going to do is we're going to simplify the right hand side of this equation. In order to simplify, we need to foil the groups A T plus B and T squared plus seven together, this is going to give us the quantity of A T cubed plus seven A T plus BT squared plus seven B. And in order to simplify the last term, we're just going to distribute T squared into C T plus D and that is going to give us C T cubed plus D T squared. Now, at this point, I recommend grouping up all the T cubed T squared T and constant values together. This is going to give us A T cubed plus C T cubed plus BT squared plus D T squared plus seven A T plus seven B. Now, what we can go ahead and do is we can factor out A T cube from the first two terms, factor out T squared from the next two terms, factor out A T from the second and last term and just leave the constant by itself. This is going to allow us to rewrite the equation as negative nine is equal to T cubed T cubed times A plus C plus T squared times B plus D plus T times seven A plus seven B. Now, this is the hard part about using partial fraction decomposition. We're going to use the simplified form of the equation to set up a system of equations that allows us to solve for ABC and D. So what we're going to do is we're going to allow the coefficient of T cubed on the right hand side, which in this case is A plus C equal to the coefficient of T cubed on the left hand side. But notice that on the left hand side, we don't have any T Q turn. So that means that T cubed on the left is going to have a coefficient of zero. So we can go ahead and set up the first equation as A plus C is equal to zero. And we're going to repeat this process for the T square terms, we have B plus DS the coefficient on the right hand side. But we have no T square term on the left hand side. That means that T squared is going to have a coefficient of zero. And we can set up the equation B plus D is equal to zero. Now, for the T terms, we only have one term which is seven A. And since there's no T term on the left hand side, that means T has a coefficient of zero. So we have seven A is equal to zero. And now we want to set the constant on the right hand side equal to the constant on the left hand side. So that's going to give us another equation which is going to be seven B is equal to negative nine. So now what we have here is a system of four equations. If we take a look at equation three, we can divide both sides of this equation by seven to give us A is equal to zero. That is going to be the first coefficient for the partial fraction decomposition. And since we have a value for A, we can go ahead and plug it into equation one to solve for C, that is just going to give us zero plus C is equal to zero, which is going to simplify to C is equal to zero, which is going to be the second coefficient. Now looking at equation four, we can divide both sides of this equation by seven to give us A B is equal to negative 9/7. This is going to be a third coefficient for the partial fractions. And since we have a B value, we can go ahead and plug this into two to give us negative 9/ plus D is equal to zero. In order to solve for D, we're just going to add 9/7 to both sides. And that is going to give us D is equal to 9/7, which is going to be the last coefficient in the partial fraction decomposition. So now that we have the coefficients for ABC and D, we're gonna go ahead and plug it back into our partial fractions. Now recall that our partial fractions was A T plus B over T squared plus C T plus D over T squared plus seven. Now keep in mind that we have A N C as zero. So if we plug in A N C, these terms are just going to cancel. L B is equal to negative 9/7 and D is equal to negative nine, I mean positive 9/7. So plugging these values in is going to give us the partial fraction of negative 9/7 over T squared plus D, which is going to be positive 9/7 over T squared plus seven. Now a shortcut to simplifying the numerator is to just go ahead and move the denominator that's in the numerator to the overall denominator. If we do this to both of the quantities, we get negative 9/7 T squared plus nine over the quantity seven times T squared plus seven. This is going to be the simplified form of the partial fractions, which means that this is going to be the simplified form of the original rational expression. And with that being said, the answer to this problem is going to be C so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-3)/(x^2(x^2 + 5))

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Polynomial Factorization

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