Question

Express in partial fraction
9/(x-1)(x+2)^2
@mathmale

Answer 1

Here, Eric, youi have a "repeated root," because the factor (x+2) in the denom. shows up twice. Repeated roots need special treatment. Let's begin with your \[9/(x-1)(x+2)^2\]and re-write it as\[\frac{ 9 }{(x-1)(x+2)^2 }\] (which makes it a bit easier to read and work with).

We want to expand this in partial fractions. First, let's do it; then let's discuss any questions about the process you may have.\[\frac{ 9 }{(x-1)(x+2)^2 }=\frac{ A }{ x-1 }+\frac{ B }{ x+2 }+ \frac{ C }{ (x+2)^2 }\]

Answer 2

Note how I've treated the single (non repeated) factor differently from the repeated factor (x+2)^2?

Answer 3

Your turn. What do you need to know or to understand better? What's the next step?

Answer 4

How to get B/x+2....?

Answer 5

@mathmale

Answer 6

Is it because of (x+2)^2 ?

Answer 7

Yes. If you have the factor (x-1) ... the same thing as (x-1)^1 ... , you'll need only one partial fraction for that factor. On the other hand, if you have the repeated factor (x+2)^2, you must write TWO partial fractions, as indicated earlier:\[\frac{ B }{ (x+2) }+\frac{ C }{ (x+2)^2 }\]

Answer 8

So, overall, your initial rational function, written in partial fractions, becomes\[\frac{ 9 }{(x-1)(x+2)^2 }=\frac{ A }{ x-1 }+\frac{ B }{ x+2 }+ \frac{ C }{ (x+2)^2 }\]

Answer 9

Do you know what to do next, Eric? Obviously you must find the values of A, B and C. How?

Answer 10

I need to make the denominator to be the same

Answer 11

Yes, that is one method, and one worth using now. Can you identify the LCD and then multiply each term of the partial fraction expansion by the appropriate quantity so that you can completely eliminate the fractions?

Answer 12

I'm weak at this part

Answer 13

\[\frac{ 9 }{(x-1)(x+2)^2 }=\frac{ A }{ x-1 }+\frac{ B }{ x+2 }+ \frac{ C }{ (x+2)^2 }\] does have an easily identifiable LCD: (x-1)(x+2)^2.

Answer 14

Now look at the first term on the right. Which factor(s) is/are missing from its denominator? To be more explicit: Look at A / (x-1) and multiply numerator and denominator both by the factor missing from the its denominator. Try that now.

Answer 15

(x+2)^2

Answer 16

that's the missing factor. yes. you must now multiply numerator and denom. of A / (x-1) by ((x+2)^2. Do that now, please.

Answer 17

A(x+2)^2/(x+1)(x+2)^2
Like this

Answer 18

Good. Now move on to the 2nd term, the one with B in the numerator. Which factor is missing from the denominator? Fix it.

Next, look at the 3rd term, the one with C in the numerator. Fix it similarly.

Type out your entire resulting expression, one that has the same denom. in all terms.

If you can use Equation Editor, that'd be great; if not, perhaps I could give you some pointers on getting started.

Answer 19

Missing factor for B is (x-1) and (x+2)
For C ... (x-1)
Is it so

Answer 20

I'd prefer you actually type out, or draw in the Draw utility, what you already have; then I'd be in a better position to give you feedback. Here's what you had before starting on the 2nd term:\[\frac{ 9 }{(x-1)(x+2)^2 }=\frac{ A }{ (x-1)(x+2)^2 }+\frac{ B }{ x+2 }+ \frac{ C }{ (x+2)^2 }\]

Answer 21

Again, the task before you is to obtain the LCD in both of the right-most two terms.

Answer 22

I'll type it in the Eqn Editor
Jst want to confirm with you
Am I correct in identifying the mising factor for both B & C ?

Answer 23

I don't see yet what you've done, so can't tell you yet whether you're correct or not.

Oh... now I understand yur question. Yes, you are to determine the factor missing from EACH denominator and then multiply both numerator and denominator by that missing factor, so that you'll end up with the same denom. as in the 1st and 2nd terms of your equation. Go to it, please.

Answer 24

\[\frac{ 9 }{ (x-1)(x+2)^2 } = \frac{ A }{ (x-1)((x+2)^2 } + \frac{ B }{ (x+2)(x+2)(x-1) } + \frac{ C }{ (x+2)^2(x-1) }\]

Answer 25

Very nice, except that for each of the 3 fractions on the right side of this equation, you must multiply BOTH numerator and denominator by the same quantity. In the case of the fraction that sharts with A, multiply both numerator and denom. by (x+2)^2.
finish the last two terms in the same manner.

Answer 26

last two terms?