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. 9x+21/(x² + 2x - 15)

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, We have four Y plus 16 divided by y squared plus four Y minus 12. And we are asked to express this in its partial fraction decomposition. Alright, we we recall from previous lessons on partial fraction decomposition that we're going to take a look at the denominator and we noticed that this denominator is not yet factored. So the first step is to factor it. So we ask ourselves what times what gives us why squared? And that's why times why? And then what? Two numbers when multiplied, give us a negative 12 but add up to a positive four and that's going to be plus six and minus two. So we can rewrite this expression as for Y plus 16 divided by R two factors why? Plus six times why minus two. And now we ask ourselves several questions about these factors in the denominator. First question are these factors linear or quadratic? And they both have a degree of one. Those wise have a degree of one. So they're both linear factors. Our next question is, are these distinct or repeating? Well, they're different from each other so they are distinct. And the last question which we've already answered, how many factors are there? There are two. And because there are two, we are going to set this fraction equal to two new fractions added together. The denominators for our two new fractions are going to be the factors from the left hand side of the equal sign denominator. So the first fractions denominator is the factor why plus six. And the second fractions denominator is the second factor why minus two? Because these are both linear factors, both of the numerator are going to be constant terms. So the numerator for the first fraction is the constant term capital A. And the second fractions numerator is capital B. And now all we have to do is go through and solve for A and B and then fill that back in up here. And this will be our partial fraction decomposition. The first step solving for A and B is to get rid of these fractions. And we're going to do that by multiplying all three of these fractions by the lowest common denominator, which is the denominator from the left hand side of the equal sign. So multiplying everything by Why? Plus six times y minus two. So that will look like a four, Y plus 16 times Why? Plus six times Y -2. All over, Y plus six times Y minus two. Equal to a capital A times Y plus six times Y minus two. All over Y plus six plus a capital B times Y plus six times y minus two. All over. Why? Minus two. And now the best part every fraction that has a factor in both. Its numerator and denominator has those factors cancel out. So on the left hand side of the equal sign, The Y plus six is on the y minus twos cancel out first fraction on the right hand side of the equal sign, The y plus six is cancel out. And the second fraction the y minus twos cancel out. So now this will read four, Y plus 16 equals capital A times y minus two plus capital B. Times Y plus six. And now we can distribute the A. And the B. So bringing down the left hand side of the equal sign equals a Y minus two. A. And distributing the B plus B, Y plus six B. And now I'd like to regroup this so that the Y terms are together and our constant terms are together. So the left hand side still says four, Y plus 16 and that equals a, Y plus B, Y minus to a plus six B. And the last part of this step is I would like to factor out A Y from R two Y terms. So this will be four, Y plus 16 equals Y times capital A plus capital B minus to a plus six B. Alright, we recall from previous lessons on partial fraction decomposition that the coefficients of like terms on either side of the equal sign are going to be equal to each other. So the coefficient of why, on the right hand side of the equal sign is a plus B. An a plus B is equal to the coefficient of y. On the left hand side of the equal sign, or it's equal to four. And likewise with our constant terms. The constant term on the right hand side is negative two A plus six B. And that is equal to the constant term on the left hand side of the equation which is 16. And now we have set up two equations and we can use this system of equations to solve for A. And B, and then go back and fill those in for our partial fraction decomposition. I would like to solve for A. And B by multiplying the first equation by two and then we'll get our A terms to cancel out. So two times A is two, A plus two times B is two, B equals four times two, which is eight. Now adding these two together are a terms cancel out, and we have eight B equals 16 plus eight is 24 Dividing both sides by eight and 24, divided by eight B equals three. Great. We've solved for B. Now filling B back in to our first equation here, equation one will read A plus not B, but A plus three equals four, and we can subtract three from both sides and we get A equals four minus three is one. So we have solved for A and B. Now we can go back up here and fill that in for our partial fraction decomposition, starting with A. So instead of a divided by one, Y plus six, it's going to be one divided by Y plus six, and for B, we're going to fill that in with three, so one divided by Y plus six plus three, divided by y minus two. We look at our answer choices and this matches with answer choice D. Well done, we'll catch you on the next one.

In Exercises 9–42, write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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