Paper 4, Section I, 2E2 E

Numbers and Sets | Part IA, 2010

(a) Let rr be a real root of the polynomial f(x)=xn+an1xn1++a0f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}, with integer coefficients aia_{i} and leading coefficient 1 . Show that if rr is rational, then rr is an integer.

(b) Write down a series for ee. By considering q!eq ! e for every natural number qq, show that ee is irrational.

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