Simplifying $\frac1{a(a-b)(a-c)}+\frac1{b(b-c)(b-a)}+\frac1{c(c-a)(c-b)}$ without opening brackets using interpolation theorems

Question

I have such expression: $$\frac{1}{a(a - b)(a-c)} + \frac{1}{b(b-c)(b - a)} + \frac{1}{c(c - a)(c-b)}$$

I need to use interpolation in order to simplify it and I basically have no idea how.

Answer 1

With $P(x)=x(x-a)(x-b)(x-c)$, you can use a decomposition of $1/P(x)$ in simple fractions : $$\frac{1}{P(x)}=\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}+\frac{D}{x}$$ where $A,B,C$ are the terms in your expression (for example, to find $A$, you multiply by $x-a$, simplify and take $x=a$), and $D=-1/abc$. Then multiply by $x$ and let $x\to \infty$ and we are done.