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

Question

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

Accepted Answer

Hello, today we're gonna be using partial fraction decomposition in order to simplify the following rational expression. So what we are given is 8/5 P times the quantity seven P plus three. Now, in order to use partial fraction decomposition, we always want to make sure that our denominator is fully factored. Currently, the given denominator is in a factored form. So we can go ahead and set up a partial fraction right away. So we're going to set up a partial fraction for each factor of P in the denominator. So the first factor of P is going to be five P. This is a non repeating linear quantity and that is going to give us the partial fraction of A over five P. Then we're gonna repeat the process for the factor of seven P plus three, seven P plus three is a non repeating linear factor. And that is going to give us a second partial fraction of B over seven P plus three. So now what we want to go ahead and do is we want to solve for coefficients of A and B. In order to do that, we want to simplify the equation we just came up with and we can simplify it by multiplying everything by a lowest common denominator. The lowest common denominator in this case is going to be five P times the quantity seven P plus three. We're gonna multiply every term by this L CD. And that is going to give us eight over five P times seven P plus three, multiplied by five P times seven P plus 3/1. And that is going to equal to a over five P times five P over seven P plus 3/ plus B over seven P plus three times five P times seven P plus 3/1. Now, we can use cross reduction in order to eliminate the denominators. In the first term, we have five P, five P times seven P plus three in both the, the numerator and denominator. So we can just go ahead and cancel out those quantities. In the second term, we have five P in both the numerator and denominator. So we can just go ahead and cancel those out. And in the last term, we have seven P plus three in both numerator and denominator. So that's just going to cancel out. That is going to leave us with the equation of eight is equal to A times seven P plus three plus B times five P. We're gonna go ahead and simplify the right hand side of this equation by distributing A into seven P plus three and multiplying B with five P that is going to give us seven A P plus three A plus five B P. Now, in this step, I'm gonna go ahead and reorder the terms. So that way all the P terms can be together and all the constants can be together. That is going to give me seven A P plus five B P plus three A. Now, what I can go ahead and do is factor out A P from the first two terms and group up the constants together. That is going to give me the equation of eight is equal to P times seven A plus five B plus the constant, which is three A. Now, this is where the tricky part of partial fractions comes in. See, we want to use this simplified form of the equation to set up a system of equations to help us solve for A and B. Well, in order to do that, we're going to let the coefficients of P on the right hand side, which in this case is seven A plus five B equal to the coefficients of P on the left hand side. But notice that on the left hand side, we don't have a P term. That means that P has a coefficient of zero. And that means that the first equation we can set up is seven A plus five B is equal to zero. Then we're going to repeat this process for our constants. So we have three A on the right hand side and we have eight on the left hand side. So the second equation we can make up is three A is equal to eight. Now we're going to sell for A and B in the second equation. If we divide three to both sides of the equation, we get A's equal to 8/3, that is going to be the first constant that we need. And since we have a value for A, we can plug this into the first equation and that is going to give us seven times 8/3 plus five. B is equal to zero seven times 8/3 is going to give us 56 over three plus five. B is equal to zero. We can subtract 50 56/3 to both sides of the equation to give us five B is equal to negative 56/3. And if we divide both sides of this equation by five, we're going to get the term that B is equal to negative 56/15, which is going to be the second coefficient that we need. So now that we have the values of A and B, we can go ahead and plug this in to the partial fractions that we set up in the beginning. Now keep in mind that the partial fractions we made was A over five P plus B over seven P plus three. Plugging an AM B into the partial fractions is going to give us 8/3, over five P minus 56/15, all over seven P plus three. So what we have is a difference of complex fractions. A shortcut into simplifying these is that you can take the denominator in the overall numerator and move it to the overall denominator. And if we do this to both of the partial fractions, we can rewrite them as 8/3 times five P minus 56/ times seven P plus three. And on the left partial fraction, we can multiply three and five P together to give us 15 P. So that means that the simplified form of the rational expression is going to be 8/15 P minus 56/15 times seven P plus three. With that being said, the answer to this problem is going to be D 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. 5/(3x(2x + 1))

Key Concepts

Partial Fraction Decomposition

Factorization of Denominators

Setting Up and Solving Equations for Coefficients

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