is it possible to trasform this expression:
a,b,c constant and x a variable. $\frac{1}{(x-a)^2}+\frac{1}{(x-b)^2}+\frac{1}{(x-f)^2}$
in a form that has the variable x outside the the parenthesis:
something like this: $(\cdots)x^4+ (\cdots)\frac{1}{x^4}+ (\cdots)x^3 + \cdots$
I'm not looking for a series expantion, not an approximation, just to have somethin with all the constant coefficents that multiply or divide x.
Is not an exercise, is that I'm not very good in math and I cannot find a way to simplify/expand this function.
for example this (expanded form): $\tiny\frac{(a^2 b^2-2 a^2 b x+a^2 f^2-2 a^2 f x+2 a^2 x^2-2 a b^2 x+4 a b x^2-2 a f^2 x+4 a f x^2-4 a x^3+b^2 f^2-2 b^2 f x+2 b^2 x^2-2 b f^2 x+4 b f x^2-4 b x^3+2 f^2 x^2-4 f x^3+3 x^4)}{((a-x)^2 (x-b)^2 (x-f)^2)}$
is not what I'm looking for.
What I'm asking first of all is: is possible? it can be that is not even possible.
What I want to do is pretty simple: I want to simulate a vector field to simulate the behaviour of n particles. The particles interacts with the others by this formula $\frac{1}{(\text{PositionParticle}_x-\text{PositionParticle}_{nx})^2}$ $\frac{1}{(\text{PositionParticle}_y-\text{PositionParticle}_{ny})^2}$ which mean that the vector field function is: $\small f(x) = \frac{1}{(\text{PositionParticle}_x-\text{PositionParticle}_{n0})^2}+\frac{1}{(\text{PositionParticle}_x-\text{PositionParticle}_{n1})^2}+(\cdots)$ same for $y$
computing this require a lot of calculation why the computer must access the array where there are stored the variable $n\times n$ times. a faster way to do this would be to calculate the vector field having the coefficients summarized and stored and then passing them to modify the $x$ variable.