Partial fraction decomposition

[dnt]

can someone help me set up this problem. it asks for the partial fraction decomposition of:

(7x^3 - 2)/[(x^2)(x+1)^3]

i thought you put A/x^2 + B/(x+1)^3 and solve but it doesn't work that way.

 

[TD]

It's not that simple: when your factors in the denominator are raised to a power n, you need n instances of that factor in your decomposition proposal (one for each exponent, 1 -> n). In your case:

$$\frac{{7x^3 - 2}}{{x^2 \left( {x + 1} \right)^3 }} = \frac{A}{{x^2 }} + \frac{B}{x} + \frac{C}{{\left( {x + 1} \right)^3 }} + \frac{D}{{\left( {x + 1} \right)^2 }} + \frac{E}{{x + 1}}$$

Do you get the idea?

 

[dnt]

yeah that makes sense. is there another way to do it where you put Ax+B over the x^2 and Cx^2 + Dx + E over the (x+1)^3 and do it with just two fractions?

 

[TD]

Sure, since (my A and B are the other way arround though)

$$\frac{A}{{x^2 }} + \frac{B}{x} = \frac{A}{{x^2 }} + \frac{{Bx}}{{x^2 }} = \frac{{A + Bx}}{{x^2 }}$$

I don't see how this would make any significant difference though...

 

[dnt]

ok cool - I am sure it wouldn't but i just wanted to be sure.