Problem #PRU-61013

Problems Algebra Polynomials Polynomials with integer coefficients and integer values Calculus Real numbers Rational and irrational numbers Number Theory Divisibility The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

13–15 3.0

Problem

Prove that if $(p, q) = 1$ and $p / q$ is a rational root of the polynomial $P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ with integer coefficients, then

a) $a_{0}$ is divisible by $p$ ;

b) $a_{n}$ is divisible by $q$ .

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