Expand the quotients in Exercises 1–8 by partial fractions.

Question

z / (z³ - z² - 6z)

Accepted Answer

Welcome back everyone. Expand x + 7 divided by x cubed minus x2 minus 2x by partial fractions. For this problem, let's go ahead and factor the denominator. We're going to have x + 7 divided by. First of all, we can factor out x, which gives us x multiplied by x 2 minus. X -2, right? So this is going to be our denominator. Additionally, we can factor out x2 minus X minus 2. Let's go ahead and do that, which gives us x + 7 divided by x, multiplied by x minus 2, which is multiplied by x + 1. Well, then, so now we have 3 linear factors, right? And what we're going to do is simply write our expression as a divided by the first linear factor x plus b divided by the second factor x minus 2 plus c divided by x + 1. And now what we can do is simply find the common denominator. Which will give us our combined numerator of a multiplied by. X minus 2. Multiplied by x plus 1. Plus b multiplied by x. Multiplied by x plus 1. Plus C multiplied by x multiplied by x minus 2. And we know that this must be equal to the original numerator of x + 7, right? We can now pick specific values of X that make the terms drop out. So first of all, we can set. X equals 0 because that will give us b multiplied by 0 and c multiplied by 0, right? If x is equal to 0, we're going to get a multiplied by. 0 minus 2, that's -2 multiplied by 0 + 1, that's 1. On the right hand side we're going to have 0 + 7, which becomes 7. So we got 2a equals 7 and therefore a is going to be equal to -7 divided by 2. Well then, this is how we get a. Now how do we get the remaining values? Well, number 2, we can set x equals 2, which will eliminate a. Right and see. Because they contain factors x minus 2. If we set x equals 2. We're going to get. B multiplied by 2 multiplied by 2 + 1, which is 3. On the right hand side we're going to have 2 + 7, which is 9. So we got 6B equals 9. And therefore B is going to be 9 divided by 6. Which is Basically, 3 halves. We can simplify dividing both sides by 3. And now let's identify the value of c. We're going to set x equals -1. And essentially. We can cancel out A and B terms because they contain a factor x + 1. Which gives us C multiplied by -1, multiplied by -1 minus 2, which is -3. And this is going to be -1 plus. 7, which is 6. Now we get 3 multiplied by C. Equals 6. So C is equal to 6 divided by 3, which is 2. Well then, and now we can use our original expression. We get a divided by X, so that's -7 halves divided by X or -7 divided by 2 X. Plus B divided by x minus 2. So that's 3 divided by 2 divided by X minus 2 or 3 divided by. 2 x minus 2. Plus C divided by x + 1 and we know that c is equal to 2, right? So we get + 2 divided by x + 1. Well then, this is going to be our final answer for this problem, and thank you for watching.

Expand the quotients in Exercises 1–8 by partial fractions.z / (z³ - z² - 6z)

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Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)