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Roots of polynomial covered by round disk

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June 4, 2019
algebrapolynomial

Problem Statement

Let P(z)P(z) be a polynomial with real coefficients whose roots are covered by a disk of radius R. Prove that for any real number kk, the roots of the polynomial nP(z)−kP′(z)nP(z)-kP'(z) can be covered by a disk of radius R+∣k∣R+|k|, where nn is the degree of P(z)P(z), and P′(z)P'(z) is the derivative of P(z)P(z).
can anyone help me? It would also be extremely helpful if anyone could tell me where they've seen this type of problems.............Has it appeared in any mathematics competitions? Or are there any similar questions for me to attempt? Thanks in advance!
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