Partial Fractions problem, need some guidance.

[DrummingAtom]

Homework Statement

$$\frac{gx^2+1}{x^3(x-1)^2}$$

I'm trying to keep g as a coefficient, it's turning into a mess though.

The Attempt at a Solution

After I broke it down it gives:

$$Ax^2(x-1)^2 + Bx(x-1)^2 + C(x-1)^2 + Dx^3(x-1) + Ex^3 = gx^2 + 1$$

After using x =1, everything goes away and I'm left with E = g + 1.

Then I used x = 0, which gave C = 1.

After that, I'm stuck. At this point I wish I knew more about linear algebra. Is there anyway to do this without going nuts with matrices? Thanks for any help.

 

[eumyang]

I would multiply all of the polynomials out and equate the coefficients on both sides.

For instance, after multiplying all of the polynomials, you will see an Ax⁴ term and a Dx⁴ term. Since there is no x⁴ term on the RHS you can say that
A + D = 0.

You'll come up with four more equations, one of which you already got (C = 1). You can then find the remaining values of A, B, D, and E via substitution. No matrices needed!