Paper 4, Section I, $2 E$

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

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

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