Equation and polynomial a) For $a,\,b,\,c\in \mathbb{R}$ , let $f(x)=x^3+ax^2+bx+c$ and $M=\max\{1,|a|+|b|+|c|\}$. Show that $f(x)>0$ for $x>M$ and $f(x)<0$ for $x<-M$
b) Consider the following polynomial with integer coefficients $a_1,...,a_n$: $P(x)=x^n+a_1 x^{n-1}+...+a_n$. Show that every rational root of $P$ is an integer.

For the problem b) first I consider it is not true that $\frac{p}{q}:(p,q)=1$ is a root of this polynomial and putting this equation $P(x)=0$ and then contradict that $(p,q)\ne 1$.
But what about a)?? Any help...

Not a rigorous proof:
$x^3 > ax^2+ bx + c$ for some $x > r$
Consider the scenario where $r$ will be greatest. Then
$f(x) = x^3 - |a|x^2 - |b|x - |c|$
For $x = 1$, we get $f(1) = 1 - |a| - |b| - |c|$.  
So if $1 \gt |a| + |b| + |c|$, then for all $x \ge 1$, $f(x) > 0$
and if $1 \lt |a| + |b| + |c|$, then for all $x \ge |a| + |b| + |c$, $f(x) > 0$
Likewise on the negative side. 
