Help With Partial Fraction Decomposition

[theintarnets]

Homework Statement

I'm supposed to decompose 1 / x(x² + 1)²
Also, we haven't learned matrices yet so I can't use that technique to solve it.

Homework Equations

None.

The Attempt at a Solution

1 / x(x² + 1)² = A/x + (Bx + C) / (x² + 1) + (Dx + E) / (x² + 1)²

I multiplied everything by the original denominator to get this:
1 = x³(Bx + C) + x²(A + B + D) + x(C + E) + A

From that, I can tell that A = 1, and I think that the following should also be true, but I'm not 100% certain:
B + D = -1
C + E = 0
B + C = 0

So I set x = -1 which gives me
B - C + 2A + B + D - C - E = 1

And I know that B + D = -1, so I can write:
B - 2C + 2A - 1 - E = 1, and since 2A is just 2, I can rewrite everything as
B - 2C - E = 0
So I take that and add it to my other equation C + E = 0 to cancel out E, and I get
B - C = 0, and then add that to my other equation B + C = 0 and then I get 2B = 0, or just
B = 0, which means D = -1 and C = 0 and E = 0
My final answer would then be
1 / x + x / (x² + 1) - x / (x² + 1)²

But that's wrong, because in the answer, only 1 / x is positive. What did I do wrong?

 

[Bread18]

i multiplied everything by the original denominator to get this:
1 = x³(bx + c) + x²(a + b + d) + x(c + e) + a

This is where your error is.
You should get, by multiplying everything by $x(x^2+1)^2$
$a(x^2+1)^2 + (bx+c)(x^2+1)x + (dx+e)x = 1$

From there expand, equate coefficients etc, and you'll get the right answer.

 

[theintarnets]

Ohhhhhhh I see, thank you!

 

[NascentOxygen]

1 / x(x2 + 1)2 = A/x + (Bx + C) / (x2 + 1) + (Dx + E) / (x2 + 1)2

Anyone know how to get wolframalpha to solve this with least effort? Would be handy to check the answer.

 

[scurty]

Anyone know how to get wolframalpha to solve this with least effort? Would be handy to check the answer.

Just type in the left hand side and it automatically does a partial fractions decomposition for you (4th box):

http://www.wolframalpha.com/input/?i=1/(x(x^2+1)^2)

 

[NascentOxygen]

Just type in the left hand side and it automatically does a partial fractions decomposition for you (4th box):

http://www.wolframalpha.com/input/?i=1/(x(x^2+1)^2)

Marvellous!