Problem #PRU-61019

Problems Calculus Derivative Derivative and multiple roots Algebra Polynomials Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas.
 14–16  3.0

Problem

For a given polynomial \(P (x)\) we describe a method that allows us to construct a polynomial \(R (x)\) that has the same roots as \(P (x)\), but all multiplicities of 1. Set \(Q (x) = (P(x), P'(x))\) and \(R (x) = P (x) Q^{-1} (x)\). Prove that

a) all the roots of the polynomial \(P (x)\) are the roots of \(R (x)\);

b) the polynomial \(R (x)\) has no multiple roots.