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. (4x^2+3x+14)/(x^3 - 8)

Accepted Answer

Hello. Today we're going to be performing partial fraction decomposition on the given expression. So the given expression is three C squared minus 10 C plus 50 over C cubed minus 1 25. Now, before we start, our partial fraction decomposition, we always want to make sure that our denominator is fully factored C cubed minus 1 25 is a factory ble quantity and is a difference of cube terms. So factoring C cubed minus 1 25 is going to give us the quantity C minus five times C squared plus five C plus 25. And now we can go ahead and set up our partial fraction decomposition. So in order to set up our partial fractions, we need to pay attention to the factors of the denominator, C minus five is a linear factor and it's non repeating. So our first partial fraction is going to be a over c minus five and now we can set up another partial fraction for our other factor C squared plus five C plus 25 is a quadratic quantity, as the leading coefficient has an exponent of two. So it's a non repeating quadratic quantity. So our second partial fraction is going to be B C b, C plus D over c squared plus five C plus 25. And now what we want to go ahead and do is solve for our unknown coefficients, A B and D. However, we can't do that unless we have a way to get rid of all these denominators while we can multiply this entire equation by the lowest common denominator and the lowest common denominator is going to be the product of all the possible denominators. So the L. C. D. Is going to be c minus five times the quantity C squared plus five C plus 25. So let's go ahead and just write that out. We have three C squared minus 10 C plus Over C -5 times C squared plus five C plus 25. And we are multiplying this quantity by the lowest common denominator, C minus five times C squared plus five C plus 25. And we're going to be doing the same thing To the partial fractions. So we're gonna multiply a over C - By C -5 times C squared plus five C plus 25. And we're gonna be multiplying B. C plus D. Over C squared plus five C plus 25 by the L. C. D. Okay, multiplying three C squared minus 10. C plus 50 is going to cancel out the L. C. D. Complete the denominator completely. And multiplying our first partial fraction by the L. C. D. Is going to leave us with a times the quantity C squared plus five C plus 25. And multiplying our second partial fraction is going to cancel our denominator, leaving us with B. C plus D. Times the quantity C minus five. So writing this out what we have is three C squared minus 10 C plus 50 is equal to a times the quantity C squared plus five C plus 25. And that's plus the quantity B C plus D times c minus five. And now what we want to go ahead and do is to simplify the right hand side. And in order to do that we're going to distribute our a into this trin Amiel and we're going to foil BC plus D with C -5. So what we have is three C squared minus 10 C plus 50. And distributing a into this. Try no meal is going to give us a C squared plus five A C Plus 25 a. And we're going to be adding B C plus D. Times c minus five. But foiling that quantity is going to give us B c squared -5 BC plus D c minus five D. Okay, this is starting to look a little bit messy but what we want to go ahead and do is combine all of the like terms together. So we're going to combine all of our c squared terms together, all of our C terms together. And all of our constant terms together. So again we're going to write down three C squared minus 10 C plus 50. But grouping up the like terms is going to give us a c squared plus B C squared plus five A c -5 BC plus D. C. And then our constants which is plus 25 A minus five D. So we've grouped up R C squared terms are see terms and our constant terms. Now, what we want to go ahead and do is factor out the common factors from each group, from r c squared terms. We can factor out C squared and from R C terms we can factor out. See So again we have the left hand side which is three, C squared minus 10, C plus 50. But factoring out the common terms is going to give us c squared times A plus B plus C times five A minus five B plus D. Plus Our Constants, which is 25 a -50. Alright, so the reason why we do this is because now we're gonna use this to set up a system of equations. That's gonna let a self for a B and D. And so what we can go ahead and do is set the factor or the coefficients of C square, which is a plus b and three equal to each other. So the first equation that we get is a plus B is equal to three and we're going to repeat this process for constant terms. And for c terms for the coefficient of C, we have five a minus five B plus D. And on the left hand side the coefficient c is negative 10. So the second equation we get is five a minus five B Plus D is equal to -10. And we're going to repeat this process for constant terms, while the only constant term on the left hand side is 50 and our constant terms on the right hand side is 25 -50. So we end up with the third equation, which is going to give us 25 a -50 is equal to 50. And so now what you want to go ahead and do is solve this system of equations to get our A. B and D. Now at this point we should be comfortable with solving a system of equations. And if you solve the system of equations you should get the coefficients of A is equal to one, B is equal to two, and D is equal to negative five. And using these coefficients, we want to go ahead and plug these into our original partial fraction decomposition, which is going to be up here. So we know that A is equal to one. So we can go ahead and replace A with one. Next we know that B is equal to two, so we can go ahead and replace B with two and finally we know that D is equal to negative five, so D Will be replaced with -5. And that means that our total partial fraction decomposition is one over c minus five plus two, C minus five over c squared plus five C plus 25. And that means that the answer to this problem is going to be A. So I hope this video helps you in understanding how to do 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. (4x^2+3x+14)/(x^3 - 8)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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