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ISI 2024 P5

Source: college entrance

May 12, 2024
limitlhopitalpolynomial limitcalculusISI 2024

Problem Statement

Let P(x)P(x) be a polynomial with real coefficients. Let α1,…,αk\alpha_1 , \dots , \alpha_k be the distinct real roots of P(x)=0P(x)=0. If P′P' is the derivative of PP, show that for each i=1,…,ki=1,\dots , k lim⁡x→αi(x−αi)P′(x)P(x)=ri,\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i, for some positive integer rir_i.
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