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Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra 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 $P (x) = (q x - p) Q (x)$ , where the polynomial $Q (x)$ also has integer coefficients.

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

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 $q$ .

Problem #PRU-98286

Problems Algebra Polynomials Polynomials with integer coefficients and integer values Calculus Real numbers Rational and irrational numbers

13–17 4.0

Let $n$ numbers are given together with their product $p$ . The difference between $p$ and each of these numbers is an odd number.

Prove that all $n$ numbers are irrational.