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. (x^3-4x^2+9x-5)/(x^2 -2x+3)^2

Accepted Answer

Hello. Today we're going to be performing partial fraction decomposition on the given expression. So the expression given to us is y cubed minus 11, Y squared plus 36 Y minus over the quantity y squared minus four Y plus five squared. Now, before we start our partial fraction decomposition, we always want to make sure that our denominator is fully factored, Y squared minus four, Y plus five is not factory bubble. So our current denominator is fully factored. However, keep in mind that y squared minus four, Y plus five is a quadratic quantity. So we have a quadratic quantity because the leading coefficient has an exponent of two and this quadratic quantity is being repeated because it's raised to the power of two. So we have a repeating quadratic quantity. And since there are two instances of this quadratic quantity, we're going to end up with two partial fractions. So the first partial fraction is going to be a Y plus b over Y squared minus four, Y plus five. And our second quadratic or second partial fraction is going to be C, y plus D over Y squared minus four, Y plus five squared. Now we want to go ahead and start solving for unknown coefficients which is a, B, C and D. But in order to do that, we need to first get rid of all of these denominators. Well, in order to do that we can multiply everything by the lowest common denominator and the lowest common denominator is going to be the highest possible exponent for the given denominators. So in this case the lowest common denominator is going to be y squared minus four. Y plus five squared. And what we're going to be doing is multiplying the L. C. D. To every term in our partial fraction equation. So just to write that out we have Y cubed minus 11 Y squared plus 36 Y minus 31 over Y squared minus four Y plus five squared. And we're going to be multiplying this by L. C. D. Which is y squared minus four Y plus five squared. And we're gonna do the same thing to our partial fractions. So we have a Y plus B over Y squared minus four. Y plus five. Multiplied by y squared minus four. Y plus five squared plus C. Y plus D. Over Y squared minus four Y plus five. That quantity squared, multiplied by the L. C. D. Which is y squared minus four Y plus five squared. Now on the left hand side multiplying this quantity by the L. C. D. Is going to completely cancel out our given denominator. And on the right hand side we have one instance of this quantity in the denominator and two in the numerator. So we're only going to cancel out the denominator but we're still left with one quantity of y squared minus four Y plus five in the numerator. And on the second partial fraction multiplying this quantity by the L. C. D. Is going to completely cancel out this denominator. So what we are left with is Y q minus 11. Y squared plus 36 Y minus is equal to a Y plus B. That quantity multiplied by y squared minus four Y plus five plus C. Y plus D. And now we want to go ahead and simplify the right hand side. Now in order to simplify this we're going to need to foil the quantity A. Y plus B. With y squared minus four Y plus five. So we have y cubed minus 11 Y squared plus 36. Y minus 31 is equal to a Y cubed minus four. A Y squared Plus five. A. Y plus B. Y squared -4. B. Y plus five B plus C. Y plus D. Now we're getting this jumble of terms here. But now what we want to go ahead and do is group up all the common terms together. So any terms that involve y cubed, we're gonna put together any terms that involve y squared, we're gonna put together any terms involved why we're gonna put together and we're gonna put all of our constants together. So doing so is going to give us y cubed minus 11 Y squared plus 36 Y minus 31 is equal to a Y cubed because there's only one cube term here plus negative four A. Y squared plus B. Y squared plus five A. Y -4 B. Y plus C. Y plus our constants Which is five B plus D. And now what we want to go ahead and do is look at our groups and factor out any common terms while in r y squared terms, both terms have the common term of y squared. So we can factor out a y squared from our first group. And on the second group we can go ahead and factor out Y. So doing this is going to give us y cubed minus 11, Y squared plus 36 Y minus 11 is equal to a Y cubed plus y squared times the quantity negative four A plus B plus y times the quantity five A minus four B plus C plus our constants, which are five B plus D. Now, what we want to go ahead and do is use this statement to set up a system of equations and use that system to solve for a B, C and D. Now, how do we do that though? But what we're gonna be doing is is equating the coefficients of our white cube terms together. The coefficients of our white square terms together, the coefficients of our white terms and the coefficients of our constants together. So the coefficients for our white cube terms is A and one because y cubed times one is just y cubed. So our first equation is going to be A is equal to one and that's one of the coefficients that we want to solve for. Now we want to go ahead and equate the y squared coefficients together. Here we have negative four A plus B and the coefficient of y squared is negative 11. So our second equation is going to be negative for a plus B is equal to negative 11. Now we want to come up with our third equation for y and we have five a minus four B plus C and the coefficient of y is stars. Third equation is going to be five, A -4. B Plus C is equal to and we have one more equation to come up with and that's just for our constant terms. So our constance put together is going to give us five B plus D equal to negative 11. So what we have here is five B plus D is equal to negative 11. Now, what you want to go ahead and do is solve the system of equations so that you can get the values for A B, C and D. And at this point we should be comfortable with solving a system of equations. So solving the system should give us the values A is equal to one, B is equal to negative seven, C is equal to three And D is equal to four. And now we want to go ahead and plug this into our original partial fractions and doing this is going to give us eight times Y or one times why? Which is why plus B, which is minus seven over the quantity Y squared minus four, Y plus five plus C. Y, which is three Y plus D, which is four over the quantity y squared minus four Y plus five squared. And this is going to be the complete partial fraction decomposition for our original expression. And that means the answer to this problem is going to be be so. I hope this video helps you in understanding how to perform a 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. (x^3-4x^2+9x-5)/(x^2 -2x+3)^2

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