The cool name is Cool Symmetry. Part III

If $A_0(x),A_1(x), \text{ and } A_2(x)$ are the three polynomials

and $a_0,a_1,\text{ and } a_2$ are three distinct real numbers, then $\dfrac{A(x)}{(x-a_0)A'(a_0)}+\dfrac{A(x)}{(x-a_1)A'(a_1)}+\dfrac{A(x)}{(x-a_2)A'(a_2)}= ?$

Note : $A'(y)$ represents the derivative of $A(x)$ at $x=y$ .

$A_0(x)=\dfrac{(x-a_1)(x-a_2)}{(a_0-a_1)(a_0-a_2)}$ $A_1(x)=\dfrac{(x-a_0)(x-a_2)}{(a_1-a_0)(a_1-a_2)}$ $A_2(x)=\dfrac{(x-a_0)(x-a_1)}{(a_2-a_0)(a_2-a_1)}$ $A(x)=(x-a_0)(x-a_1)(x-a_2)$