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. (7x-4)/(x^2-x-12)

Accepted Answer

welcome back. I am so glad you're here. We are given this expression to Why minus nine divided by. Why squared minus two Y minus 35. And we are asked to express this in its part fraction decomposition. Alright, well recalling previous lessons on partial fraction decomposition, we're going to take a look at our denominator and we notice that this denominator is not yet factored. So that's the first step. We're going to factor it. So what times what will give us Y squared? That would be a Y times Y. And what? Two numbers when multiplied, give us a negative 35. But when added together, give us a negative two and that would be a minus seven plus five. All right. So now we can rewrite this expression two. Why? Minus nine in the numerator? And are factors in the denominator. So why? Minus seven times Why? Plus five? And for partial fraction decomposition, we're going to ask ourselves several questions about these factors in the denominator. First are these linear or quadratic? And both of those wise have a degree of one. So, they are both linear factors. Next we ask ourselves if they are distinct or repeating and they're different from one another. So they are distinct. And the last question which we've already answered is how many factors are there? There are two of them. So, because there are two. We are going to set this equal to two new fractions added together. And for these two new fractions the denominators are going to be the factors from the previous fractions denominator. So the first new fraction. The denominator will be the first factor, y minus seven and the second fraction will have the second factor as its denominator, Y plus five. And because they are both linear, they're numerator are going to be constant terms. So for the first fraction we're going to have the constant term A in the numerator, and the second fraction will be the constant term B in the numerator. And now all we have to do is solve for A and B. And come back up here and fill them in. And this will be our answer in partial fraction decomposition. In order to solve for A and B. I want to start off by getting rid of our fractions. And in order to do that, we need to multiply 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 y minus seven times Y plus five. So, the way that will look, we'll have two y minus nine times Y -7 times y plus five. All over y minus seven times Y plus five, equal to capital A times y minus seven times Y plus five. All over y minus seven plus capital B times Y minus seven times Y plus five. All over Y plus five. And now the most fun part, anywhere in a fraction where they have a factor that occurs in both the numerator and the denominator, those factors cancel out. So on the left hand side of the equal sign, the y minus seven's and the y plus fives cancel out. On the right hand side of the equal sign. The first fraction the y minus seven's cancel out. And the second fraction the Y plus fives cancel out. Yeah, er, fractions are gone and we've now got two y minus nine equals capital A times Y plus five plus capital B times y minus seven. Now we can bring down the left hand side of the equal sign and distribute our capital letters on the right hand side. So distributing the A. That will be a, Y plus five A plus distributing the B, B, y minus seven B. And now I'd like to regroup these so that our Y terms are together and our constant terms are together. So bringing down the left hand side to y minus nine equals, and now are Y terms together A, Y plus B, Y. And our constant terms together plus five A minus seven B. And the last step on this part of the work, I'd like to factor out the Y from R two Y terms. So we have two y minus nine equals Y times a plus B plus five A minus seven B. Now we recall from previous lessons on partial fraction decomposition that the coefficients for our like terms on either side of the equal sign are equal to each other. So the coefficient for y on the right hand side of the equation is A plus B, and A plus B is equal to the coefficient of why from the left hand side of the equation, So A plus B is equal to two. And likewise with our constant terms are constant terms from the right hand side of the equation. five A -7 b. Those are equal to Our constant term from the left hand side of the equation, which is -9. And now we have set up two equations. And from this system of equations we can solve for A and B and then go back and fill them in for our partial fraction decomposition. Alright to solve for A and B. I would like to start off by multiplying our first equation times seven and then we'll add them together and we'll get RB terms to cancel out so that we can solve for a. So seven times a is seven, A. Seven times B is plus seven B equals seven times two is 14, adding these two equations together. Five A plus seven A is 12. A. R B terms cancel out, and that equals negative nine plus 14 is positive five. And now solving for a will divide both sides by and we get a equals 5/12. All right. We've solved for a. Now let's fill that into our first equation here will fill in 5/12 Plus B equals two and we can subtract 5/ from both sides, and we'll get B equals two minus five. Twelves is 1912. And we have solved for A and B. And now we can go back up here to our partial fraction decomposition and fill in for A and B. So instead of A over y minus seven, it will be 5/12 divided by y minus seven, and instead of B over Y plus five, it is 1912 months, divided by Y plus five. We look at our answer choices and this matches with answer choice B. Well done, we'll catch on the next one.

In Exercises 9–42, write the partial fraction decomposition of each rational expression. (7x-4)/(x^2-x-12)

Key Concepts

Rational Expressions

Partial Fraction Decomposition

Factoring Polynomials

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