i dont get this problem at all. plz help.?

Question

i first asked this question on http://answers.yahoo.com/question/index;_ylt=AjQxaxmnAQxwPQVIzFpolhnsy6IX;_ylv=3?qid=20071023083937AAfnneT

altho i do get some of the work. I dont get all of it. For example, i dont get this part.then just concentrate on the numerator...

1 = [A(x-1)(x+1) + Bx(x+1) + Cx(x-1)]

substitute x = 0 , 1 & -1to get the values of A,B & C...

at x = 0
1 = -A

at x = 1
1 = 2B

at x = -1
1 = 2C

thus
1/(x^3 - x) = (-1)/x + (1/2)/(x-1) + (1/2)/(x+1)

so what is A,B, and C, then? do i solve further for it or something? I'm trying but I am just really bad at math.
:(
so please help. Thanks

Answer 1

No, you're done. Your result,

1/(x^3 - x) = (-1)/x + (1/2)/(x-1) + (1/2)/(x+1)

is an identity (that is, it's true for all values of x for which it is defined).

The equation

1 = [A(x-1)(x+1) + Bx(x+1) + Cx(x-1)]

is actually

0x^2 + 0x + 1 = [A(x-1)(x+1) + Bx(x+1) + Cx(x-1)]

A, B, and C are unknown constants that are to reproduce the polynomial I've written on the left side. A more laborious method of solution is to multiply out the right side and match up coefficients of the different powers of x on the two sides of the equation. So, doing it this way, you would get

0x^2 + 0x + 1 = Ax^2 - A + Bx^2 + Bx + Cx^2 - Cx

or after grouping terms on the right side

0x^2 + 0x + 1 = (A+B+C)x^2 + (B-C)x - A

Matching up the coefficients of powers of x gives the equations

(from coef. of x^2) 0 = A + B + C
(from coef. of x) 0 = B - C
(from constant) 1 = -A

The solution of that system is A = -1, B = 1/2, C = 1/2, just as was found using the easier method.