Problem #PRU-32884

Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra Polynomials Polynomial remainder theorem. Factorisation. Polynomials with integer coefficients and integer values Proof by contradiction Calculus Real numbers Rational and irrational numbers
 12–12  3.0

Problem

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.