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Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation. Polynomials with integer coefficients and integer values Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–12 3.0

Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.

Problem #PRU-61018

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation.

15–16 3.0

Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).