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Find the partial fraction decomposition for $\frac{8 x^{2}+17 x+12}{x^{3}+3 x^{2}+2 x}$

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Algebra

Chapter 9

Matrices and Determinants

Section 2

Operations with Matrices

Introduction to Matrices

McMaster University

Baylor University

University of Michigan - Ann Arbor

Lectures

03:17

Find each partial fraction…

03:29

Find the partial-fraction …

06:31

Partial Fraction Decomposi…

to find the partial fraction decomposition of this rational expression. We need to start by factoring the denominator and the denominator factors into X Times X plus one times X plus two and the numerator can stay the same. So this means that our partial fractions there will be three of them, one with the denominator x one with the denominator X plus one and one with the denominator X Plus two and the numerator CZ. For now, we'll call them A B and C on our job here is to figure out the values of A, B and C. So now we have a rational equation, and we're gonna multiply both sides of the equation by X by X plus one and by X plus two. What that does to the left side of the equation is it eliminates the denominator. And now what we have on the left is eight x squared plus 17 x plus 12 and on the right. The first fraction has X on the bottom that cancels with our X that we multiplied by and we're left with a Times X plus one times X plus two. For the second fraction, the X plus ones cancel and we're left with B times X Times X plus two. And for the third fraction, the X plus twos. Cancel and we're left with C times, X Times X plus one. We're going to use this equation to find the values of A, B and C, and the method I'm going to use is to substitute inconvenient values of X. So for the first convenient value of X, I'm going to choose X equals zero. If X equals zero, the left side of the equation is just 12 and the right side of the equation will be a times one times two. So two way plus B time zero because X will be zero plus c time. Zero. So all we have is 12 equals two a. And that tells us that is going to be six. Now I'm going to find another convenient value for X to tuck to substitute in. So I'm going to use X equals negative one because it makes this factor zero and then after that, for the third convenient value of X to plug in, I'm going to choose X equals negative, too, because it makes this factor zero. Okay, so Here we go. Substituting in X equals negative one, and that's going to give us three equals. The first part will be zero because we have the X plus one there, so that's going to be zero. The next part will be be times negative, one times one. So that's gonna be the opposite of B. And the third part will be zero, because the X plus one will be zero. So now we know that B is negative three and then lastly, we can substitute in X equals negative, too. And the left side of the equation will be 10. And this part is going to be zero. So the whole first term is hero. This part is going to be zero. So the whole second term zero and the last term is going to be C times negative two times negative one. And that's to see. So now we know that sea is five. Okay, now that we know a, B and C remember that a waas six, we substitute those back into the partial fractions, and our answer is six over X minus three over X plus one minus three. Because we had negative three for B plus five over X Plus two, and you can check that answer by adding it's attracting those fractions and see that it matches with the original problem.

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