The cool name is Cool Symmetry. Part IV

Algebra Level 4

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, and if \(\dfrac{1}{A(x)}=\dfrac{A_0}{x-a_0}+\dfrac{A_1}{x-a_1}+\dfrac{A_2}{x-a_2}\), then \[A_i \text{ (for i=0,1,2) equals : } \]

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

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

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