Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The function in question that I want to decompose is $$\dfrac{8x^3 + 7}{(x+1)(2x+1)^3}$$

I had the idea to to break this down into:
$$\dfrac{A}{x+1} + \dfrac{Bx^2 +Cx + D}{(2x+1)^3} + \dfrac{Ex + F}{(2x+1)^2} + \dfrac{G}{2x+1}$$

Well this turns into a really messy system of 4 equations and 7 variables. Is there an easier way to decompose the function? Can I possibly eliminate any of these variables at the outset?

share|cite|improve this question
Here is a partial fraction technique which handles this problem. – Mhenni Benghorbal Jul 6 '13 at 9:27
up vote 3 down vote accepted

According to Partial Fraction Decomposition rule, it will be $$\frac{A}{x+1} + \frac{B}{(2x+1)^3} + \frac{E}{(2x+1)^2} + \frac{G}{2x+1}$$

share|cite|improve this answer
@Mark, could you have a look into the link in the answer or – lab bhattacharjee Feb 15 '13 at 18:33

I'll simply alert you to the existence of two separate methods for dealing with partial fractions with repeated roots that may potentially simplify the process:

1) Ostrogradsky's method

2) Hardy's Method (Page 10 on)

share|cite|improve this answer
Thanks for the links! – vonbrand Feb 15 '13 at 20:33

You only need to break it down into $\displaystyle\frac{A}{x+1}$ and $\displaystyle\frac{Bx^2 + Cx + D}{(2x+1)^3}.$ What you end up with is $\begin{align*} 8A + B &= 8\\ 12A + B +C &= 0\\ 6A + C + D &= 0\\ A + D &= 7 \end{align*}$

Solving gives $A = 1, B = 0, C = -12, D =6.$

Note that this is equivalent to lab bhattacharjee's decomposition.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.