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. x²+2x+7/x(x − 1)^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 negative two A squared plus 13 A plus 36 Over a times a -6 squared. Now, before we perform a partial fraction decomposition, we always want to check to make sure that our denominator is fully factored. A. Is just a variable on its own. So there's really nothing you can do with it and a -6 is not fact arable as it's already in its simplest terms. So our denominator is fully factored. Now let's go ahead and determine what kind of partial fractions that the factors are going to produce. A is a variable on its own. However, it's considered a linear quantity because it's a variable with an exponent of one. So the factor A is going to produce the partial fraction A over A. And a minus six is a linear quantity as its leading coefficient has an exponent of one. But this quantity is raised to the second power so it's being repeated two times. So this quantity a minus six squared is going to produce two partial fractions. The partial fractions are going to be B over a minus six plus C over a minus six squared. Now, what we want to go ahead and do is sulfur A. B and C. And in order to do that we need to first get rid of all of these denominators in order to do that we can multiply everything by the lowest common denominator. And the L. C. D. In this case is going to be a times a minus six squared. So let's go ahead and multiply everything by the L. C. D. That's going to give us negative two A squared plus 13 A plus 36 over a times a minus six squared, multiplied by the L. C. D. Which is eight times a minus six squared over one. And that is equal to our first partial fraction which is A over a times the L. C. D. Which is eight times a minus six squared over one plus our second partial fraction which is B over a minus six times the L. C. D. Which is eight times a minus six squared over one plus our third partial fraction which is C over a minus six squared times are L. C. D. Which is a times a minus six squared over one. Now multiplying the left hand side by the L. C. D. Is going to completely cancel out the denominator. And on the right hand side multiplying the first partial fraction with the L. C. D. Is going to cancel the A. Believe us with a minus six squared in the numerator. On our second partial fraction, we're gonna cancel out one instance of a minus six. So we have B times a times a minus six. And for our third partial fraction multiplying by the L. C. D. Is going to completely cancel at a minus six squared Believe us with A. In the numerator. So what we have is negative two A. Squared plus 13 A. Plus 36 is equal to a. Times a minus six squared plus B. Times A. Which is B. A. Times a minus six plus C. Times A. Which is C. A. Now what we want to go ahead and do is simplify the right hand side. And in order to do that we're going to go ahead and foil A minus six and distribute B. A. Into A minus six. And doing so is going to give us negative two A. Squared plus 13 A. Plus 36 is equal to A. Times the quantity A squared minus 12 A. Plus 36 plus B. A. Squared -6 B. A. Plus C. A. And there's one more thing we need to do. We need to go ahead and distribute A. Into the quantity A squared minus 12 A. Plus 36. That's going to give us negative two A squared plus 13 A. Plus 36 is equal to a times a squared -12 A. times a Plus 36 A plus B. A squared -6 B. A. Plus C. A. And now what we want to go ahead and do is group up all of our common terms together. So we're gonna group up all of our a square terms together all over eight terms together and all of the constant terms together and doing so is going to give us two negative two A. Squared plus 13 8 plus 36 equal to a times a squared plus B. A squared plus -12 A times a - BA plus plus C. A. And then plus 36 8. And now what we want to go ahead and do is factor out any common factors. In our first group we have a squared as a common factor and in our second group we have a as a common factor. So factoring those out is going to give us native to a squared plus 13 A plus 36 is equal to a squared times A plus B plus a. Times negative 12 A -6 B Plus C plus 36 A. And now what we want to go ahead and do is create a system of equations in order to solve for a B and C. And in order to do that, we're going to take the coefficients of a squared and set them equal to each other, take the coefficients of A and set them equal to each other and take our constants and set them equal to each other. So our first expression, our first equation is going to be the coefficients of a square which is A plus B. And that is going to equal to negative two. For our second equation, we're taking the coefficients of A and sending them equal to each other so that's gonna be negative A minus six B Plus C equal to 13. And for a third equation we're just taking our constant term and setting it equal to each other so we get 36 a is equal to 36. And we're gonna use this system of equations to solve for a B and C. But by this point we should be comfortable with solving a system of equations. So once you solve the system of equations you're going to get the coefficients of A is equal to one, B is equal to negative three and C is equal to seven. And now what you want to go ahead and do is take these coefficients and plug them into your original partial fractions. Now recall that the original partial fractions was capital A over A plus B over a minus six plus C over a minus six squared. Now replacing A with one is going to give us one over a replacing B with negative three is going to give us negative three over a minus six, but we can go ahead and move the negative to the front of the fraction. So we have minus three over a minus six and replacing C with seven is going to give us seven Over a - squared. And this is going to be the complete partial fraction decomposition for a given expression. And that means that 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. x²+2x+7/x(x − 1)^2

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Polynomial Factorization

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