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

Question

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

Accepted Answer

Hello, today we're gonna be using partial fraction decomposition to simplify the rational expression. So what we are given is P over the quantity P squared minus four P minus 21. Now, in order to use partial fraction decomposition, we always want to have our denominator in a fully factored form. Currently, our denominator is in the form of a factor trinomial. And if we factor this using the trial and error method, we can rewrite the rational expression as P over the quantity P minus seven times P plus three. Now that we have our denominator in a fully factored form, we can go ahead and make our partial fractions. So we want to make a partial fraction for every factor of P in the denominator. The first factor of P we have is P minus seven. This is a non repeating linear quantity and that is going to give us the partial fraction of A over P minus seven. Then we're going to make a second partial fraction for the other factor of P. The second factor of P is P plus three. This is also a non repeating linear linear combination or linear quantity. And that is going to give us B over P plus three. So now that we have the partial fraction equation set up, we can go ahead and use this equation to solve for the coefficients of A and B. And in order to do that, we want to multiply each term of the equation by the lowest common denominator. In order to solve for A and B. In this case here, the lowest common denominator is going to be P minus seven times P plus three. So we're going to multiply each term in this equation by the lowest common denominator. In order to simplify it, that is going to give us P over P minus seven times P plus three multiplied by the L CD, which is going to be P minus seven times P plus 3/1. And that is going to equal to A over P minus seven times the L CD, which is P minus seven times P plus 3/ plus B over P plus three times the L CD, which is P minus seven times P plus 3/1. We can go ahead and use cross cross simplification in order to eliminate the denominators. So in the first term, we have P minus seven times P plus three in both numerator and denominator. So we can go ahead and eliminate those quantities. In the second term, we have P minus seven in both the numerator and denominator. So we can go ahead and eliminate those quantities. And in the last term, we have P plus three in both numerator and denominator. So we can just go ahead and eliminate that quantity. That is going to leave us with the equation P is equal to A times the quantity P plus three plus B times the quantity P minus seven. Now, what we want to go ahead and do is we want to simplify this equation in order to do that, we're going to distribute A into P plus three and we're going to distribute B into P minus seven that is going to give us A P plus three A plus B P minus seven B. Now, at this point, I highly recommend grouping up or reordering the terms. So that way the P terms can be together and the constants can be together. So if we quickly reorder the terms on the right hand side of the equation, we get a times P plus B times P plus three A minus seven B. Now, what we can go ahead and do is we can factor out A P from the first two terms and group up the two constants together. That is going to give us the equation P is equal to P times A plus B plus the constants three A minus seven B. Now this is where the tricky part of partial fraction decomposition comes in. We want to go ahead and use this equation to set up A system of equations that's going to allow us to solve for A and B. So in order to do that, we're going to allow the coefficient of P on the right hand side, which in this case is A plus B equal to the coefficient of P on the left hand side, which is going to be one. So that's going to give us the first equation which is going to be A plus B is equal to one. Now, we want the constant on the right hand side to equal to the constant on the left hand side. But notice that we don't have any constant values on the left hand side. So that is going to give us the equation three A minus seven B is equal to zero. Now, we can go ahead and solve for A and B using the method of solving as a system of equations. So let's go ahead and eliminate the A term. We have three A in the second equation. And we have a, in the first equation. If we multiply the first equation by negative three, we can rewrite the system as negative three A minus three B is equal to negative three. And then we're gonna add the second equation which is three A minus seven B is equal to zero, negative three A plus three A is just going to cancel each other out and negative three B minus seven B is going to give us negative 10 B and negative three plus zero is just going to give us negative three. If we divide both sides of this equation by negative 10, we get the quantity B is equal to 3/10, which is going to be the first coefficient that we need. Now that we have the value of B, we can go ahead and plug this into the original form of the first equation to sulfur A. That is going to give us a plus 3/10 is equal to one. In order to solve for A, we're going to subtract 3/10 to both sides of the equation. And if we do this, this is going to give us the value of A is equal to 7/10. So now that we have the coefficients for A and B, we can go ahead and plug this back into the partial fraction decomposition. Now recall that our original partial fractions were in the form of A over P minus seven plus B over P plus three. If we plug in the values for A and B, we get 7/10 over P minus seven plus 3/10 over P plus three, this is in the form of complex fractions. A shortcut to simplifying the complex fractions is to take the denominator that's in the overall numerator and move it to the overall denominator. If we do this to both of the partial fractions, we can rewrite the partial fractions as seven over the quantity, 10 times P minus seven plus 3/10 times the quantity P plus three. And this is going to be the fully simplified form of the original rational expression. And with that being said, the answer to this problem is going to be B 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. x/(x^2 + 4x - 5)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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