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.5x^2+6x+3/(x + 1)(x² + 2x + 2)

Accepted Answer

Hello. Today we're going to be performing partial fraction decomposition on the given expression. So what we are given is six Y squared minus three. Y plus 45 over. Wide minus five times Y squared plus three Y plus five. Now, before we start, our partial fraction decomposition, we always want to make sure whether our denominator is fully factored or not. Why minus five is in its simplest form and why squared plus three Y plus five can no longer be factored. So the denominator is in its simplest form. Now let's go ahead and see what the factors produce. We have Y -5 which is a linear quantity because the leading coefficient has an exponent of one. So why -5 is going to produce the partial fraction a. Over Y -5. And when we take a look at y squared plus three Y plus five. This is a quadratic quantity because the leading coefficient has an exponent of two. So y squared plus three, Y plus five is going to produce the partial fraction B, Y plus C over Y squared plus three Y plus five. And now we want to go ahead and sulfur are a. B and C coefficients. But in order to do that, we need to first get rid of all of these denominators in order to get rid of the denominators. We're gonna multiply by the L. C. D. And the L. C. D. In this case is going to be y minus five times Y squared plus three Y plus five. So let's go ahead and write this out, we have six Y squared minus three. Y plus 45 over y minus five times Y squared plus three Y plus five. And we're multiplying that by the L. C. D. Which is y minus five times Y squared plus three Y plus five. And that's gonna be over one. And that is equal to our first partial fraction which is a over y minus five times the L. C. D. Which is why minus five times Y squared plus three Y plus 5/1 plus our second partial fraction which is B. Y. Plus C. Over Y squared plus three Y plus five times the L. C. D. Which is y minus five times Y squared plus three Y plus five. All over one. Now on the left hand side multiplying this quantity by the L. C. D. Is going to completely cancel out the denominator On our first partial fraction multiplying this quantity by the L. C. D. Is going to cancel out the denominator of y minus five. But leave us with Y squared plus three Y plus five in the numerator. And on our second partial fraction multiplying by the L. C. D. Is going to cancel out the denominator. Believe us with the quantity of why minus five in the numerator. So writing this out is going to give us 66 Y squared minus three Y plus 45 is equal to a times the quantity Y squared plus three Y plus five plus the quantity B. Y plus C Times The Quantity. Why -5. Now what we want to go ahead and do is simplify the right hand side of this of this equation. And in order to do that we're going to distribute a into the quantity Y squared plus three Y plus five. And we're going to foil B Y plus C. With y minus five. Doing this is going to give us six, Y squared minus three. Y plus 45 is equal to a Y squared plus three A Y plus five A plus B. Y squared -5, B. Y plus C, y minus five C. Now this is a mess of terms. But what we want to go ahead and do is group up all the like terms together. So we're gonna group up all the y squared terms together all the y terms together and all the constants together. And doing this is going to give us six Y squared minus three. Y plus 45 is equal to a Y squared plus B. Y squared plus three A Y -5, B. Y plus C. Y Plus our constants which is five a minus five C. And now what we're going to go ahead and do is factor out any common factors. So in our first group we have a common factor of Y squared so we can go ahead and factor out Y squared and in our second group we have a common factor of Y. So we can go ahead and factor out why And this is going to give us six, y squared minus three, Y plus 45 is equal to y squared times the quantity A plus B Plus y times the quantity three A -5 B Plus C plus our constants which is five a minus five C. And now in order to solve for a B and C, we want to go ahead and create a system of equations where we set the coefficients of y squared y and the constants equal to each other. So let's go ahead and start with the y squared coefficients on the right hand side. The coefficient of y squared is A plus B. And on the left hand side the coefficient of six of is the coefficient of y squared is six. So what we have is a plus B is equal to six. For a second equation, we're gonna set the coefficient of y on the right hand side which is three a minus five B plus C equal to the coefficient of y on the left hand side which is negative three. So what we have is three a minus five B Plus C is equal to -3. And now we want to go ahead and set our constants equal to each other for a third equation, the constants on the right hand side is five a minus five C. And the constant on the left hand side is 45. So we get the equation five a minus five, C is equal to 45. And now, once we solve the system of equations, we're gonna get our our values for a B and C. Now by this point we should be comfortable with solving a system of equations. So once you do solve this, you should get the coefficients A is equal to four, B is equal to two and C is equal to negative five. And now you're gonna want to take these and plug it into the original partial fraction decomposition. While recall that the partial fraction decomposition was a over y minus five plus B, Y plus C. Over why squared Plus three Y Plus five. So replacing A with four is going to give us four over y minus five, replacing B with two is going to give us two Y plus C. And replacing seat with -5 is going to give us two Y -5. And this is going to be the complete partial fraction decomposition for a given expression. And that means the answer to this problem is going to be C. So, I hope this video helps you in understanding how to perform 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.5x^2+6x+3/(x + 1)(x² + 2x + 2)

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