Math Help - Expand Into Partial Fractions

1. Expand Into Partial Fractions

I know how to do these problems but this one is giving me some trouble:
Expand into partial fractions

5/(x+1)(x^2-1)

***Should I break it down into:
1) [A/(x+1)]+[B/(x-1)]+[C/(x+1)] or
2) [A/(x+1)]+[B/(x+1)^2]+[C/(x-1)]

I tried it both ways and came up with different possibilities for the constraints:
If Broken Down Like #1 from Above
A+B+C=0
2B = 0
A+B-C=5

If Broken Down Like #2 from Above
A+C=0
B+2C=0
A+B-C=5

2. Re: Expand Into Partial Fractions

Which way gives you the LCD that is the denominator of the original expression?

3. Re: Expand Into Partial Fractions

I'm not sure because you can break down the original denominator (x+1)(x^2-1) into (x+1)(x+1)(x-1)

4. Re: Expand Into Partial Fractions

Yes, and if you were to combine the 3 terms of both the choices you listed, which one would give you the correct common denominator?

5. Re: Expand Into Partial Fractions

Option 1 would give me that..right?

6. Re: Expand Into Partial Fractions

No it wouldn't. Since two of them have \displaystyle \begin{align*} x + 1 \end{align*} as the denominator, the LCD is \displaystyle \begin{align*} (x + 1)(x - 1) \end{align*}, NOT \displaystyle \begin{align*} (x + 1)^2(x - 1) \end{align*} as required.

When you have repeated factors, you need one denominator to be \displaystyle \begin{align*} x + 1 \end{align*} and another to be \displaystyle \begin{align*} (x + 1)^2 \end{align*} in your partial fraction.

7. Re: Expand Into Partial Fractions

so when I write it in the partial fractions I write it as:
[A/(x+1)]+[B/(x+1)^2]+[C/(x-1)]

??

Yes

9. Re: Expand Into Partial Fractions

5/(x+1)(x^2-1) = A/(x+1) + B/(x+1)^2 + C/(x-1)
multiply by LCD to get:

5= A(x+1)(x-1) + B(x-1) + C(x+1)^2

5= A(x^2-1) + B(x-1) + C(x^2+2x+1)

5= Ax^2 - A + Bx - B + Cx^2 + 2Cx + C

Group Common Terms:
5= x^2(A+C) + x(B+2C) + (-A-B+C)

Constraints:
A+C=0
B+2C=0
A+B-C=5

10. Re: Expand Into Partial Fractions

Originally Posted by rrooggeerr
5/(x+1)(x^2-1) = A/(x+1) + B/(x+1)^2 + C/(x-1)
multiply by LCD to get:

5= A(x+1)(x-1) + B(x-1) + C(x+1)^2

5= A(x^2-1) + B(x-1) + C(x^2+2x+1)

5= Ax^2 - A + Bx - B + Cx^2 + 2Cx + C

Group Common Terms:
5= x^2(A+C) + x(B+2C) + (-A-B+C)

Constraints:
A+C=0
B+2C=0
A+B-C=5
This all looks fine. Now solve for A,B,C.

11. Re: Expand Into Partial Fractions

I get:
A= -5/4
B= -5/2
C= 5/4

I know something is wrong though because it doesn't make the third constraint true

12. Re: Expand Into Partial Fractions

never mind, i got it!!

A = 5/4
B= 5/2
2= -5/4

THANK YOU!

13. Re: Expand Into Partial Fractions

There is an alternate method for partial fraction decomposition you may be interested in:

Heaviside cover-up method - Wikipedia, the free encyclopedia