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.
Let the number \(\alpha\) be given by the decimal:
a) \(0.101001000100001000001 \dots\);
b) \(0.123456789101112131415 \dots\).
Will this number be rational?
Prove that the number \(\sqrt {2} + \sqrt {3} + \sqrt {5} + \sqrt {7} + \sqrt {11} + \sqrt {13} + \sqrt {17}\) is irrational.