# The cool name is Cool Symmetry. Part III

**Algebra**Level 4

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

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

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 \(x=y\).

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