In Exercises 16–24, write the partial fraction decomposition of e...

Question

In Exercises 16–24, write the partial fraction decomposition of each rational expression.3x/(x - 2)(x^2 + 1)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this rational expression which is two X over X plus two times X squared plus four. And we want to decompose that. And let's see what's going on with our denominator. We've got this linear factor X plus two and we've got a quadratic factor X squared plus four. That's prime. We can't factor it out anymore. And it's also non repeating. So when we've got a non repeating prime quadratic factor involved, we set this equal to a Over our linear factor which is X-plus two plus B. X plus C. Over a quadratic factor which is X squared plus four. And now it's just a matter of solving for a B and C. We're going to multiply everything by the lowest common denominator which is what's the denominator on the left side which is X plus two times X squared plus four. And on that left hand side the denominator is going to cancel out by what's being multiplied by the numerator. So it's just two X. The a term. Well the X plus two is going to cancel out the X plus two. So it's going to get multiplied by the X squared plus four and that be X plus C factor while the X squared plus fours will cancel out. And that just gets multiplied by the X plus two. Now let's distribute. So we've got a X squared plus four A. Plus now we foil, B X squared plus two, X plus C X plus two C. And now let's rewrite that with all the terms in our descending order for X. So two X equals A X squared plus bx squared. Now let's do our X terms. So plus to be X plus C, X. And now our constant terms plus four A plus two C. Great. Those are in order. And now let's factor out where we can we can factor out an X squared. We've got x squared times A plus B. Plus we can factor out an X. We've got X Times two B Plus C. And now it's like we're factoring out of one for our constant terms and we've got plus four A plus two C. And now we know that coefficients on the left will match coefficients on the right and well it's a little bit different because there is no X squared term on the right, it's like we have an zero X squared. And so the coefficient of the X squared is zero and the coefficient on the right hand side is A plus B. So I'm going to rewrite it with a plus B on the left Equals The Coefficient zero. Now let's look at our x terms and left, our coefficient is two and on the right, we've got to be plus C In our constant terms. Well, there's no constant term on the left, so again, that's like a zero And that's going to equal R four, a plus two c. And now we can just um add those terms together or add those equations together in our system of equations we have here in order to solve for A. B and C. We have a lot of different options but we could start off by multiplying this top one by negative two. And so then we'd have negative two. A. And let's add it to the second equation. So negative to a negative two times B plus two B is going to cancel out. There's no C. Term on the top one. So we're just going to bring down that plus C. And zero times negative +20 plus two is still too. So there's one of our new equations and we can add the third the purple and the fourth the black equations together. And let's multiply this new one times 2. So four A plus two times negative two. A. The A's will cancel out to C plus two times C. Well so that's another to see. So that'll be four. C equals zero. Plus two times two will be four. So now we know that C equals one. We've got one of them. We can plug that back in. Let's do it with our second equation. So we've got two B plus. Now we know that C. Is one equals to subtract one from both sides. So we've got two B equals one and that's going to give us B equals one half. Got two of them. Let's plug that into our first one. So A plus one half equals zero. Well subtract one half from both sides and A equals negative one half. We have got all three A. B and C solved and now it's just plugging these in. So A. Is negative one half up here. B is going to be one half times X plus C. Which is one for that B. Term. We know that one half X is the same thing as X over two. So we'll rewrite it like that where we've got negative one half over X plus two plus X over two plus one over X squared plus four. And this is going to match with our answer choice. A. Well done. We'll catch you on the next one.

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