First Principles AI

Solution

Let $t=x^2$ . Then:

\frac{x^2}{(x^2+a)(x^2+b)(x^2+c)}=\frac{t}{(t+a)(t+b)(t+c)}

Assume:

\frac{t}{(t+a)(t+b)(t+c)}=\frac{A}{t+a}+\frac{B}{t+b}+\frac{C}{t+c}

So:

t=A(t+b)(t+c)+B(t+a)(t+c)+C(t+a)(t+b)

Put $t=-a$ :

-a=A(b-a)(c-a)\Rightarrow A=\frac{a}{(a-b)(a-c)}

Put $t=-b$ :

-b=B(a-b)(c-b)\Rightarrow B=\frac{b}{(b-a)(b-c)}

Put $t=-c$ :

-c=C(a-c)(b-c)\Rightarrow C=\frac{c}{(c-a)(c-b)}

Now replace $t=x^2$ :

\boxed{\frac{x^2}{(x^2+a)(x^2+b)(x^2+c)}=\frac{\dfrac{a}{(a-b)(a-c)}}{x^2+a}+\frac{\dfrac{b}{(b-a)(b-c)}}{x^2+b}+\frac{\dfrac{c}{(c-a)(c-b)}}{x^2+c}}