The exact theorem I've been asked to prove is the following:
Suppose $f(x)=a_n x^n + a_{n-1}x^{n-1} + …+a_0$ is a polynomial of degree $n>0$ and suppose $a_n>0$. Then there is an integer $k$ such that if $x>k$, then $f(x)>0$.
(This is working up to the proof that any polynomial of degree $n>0$ is a composite number for infinitely many integers $x$.)
I have a suspicion that the proof I'm stuck on is extremely simple but I can't even think of where to begin.
Any help/hints would be appreciated!
Best Answer
$$f(x)=x^n\left(a_n+a_{n-1}\frac1x+a_{n-2}\frac1{x^2}+\ldots+a_1\frac1{x^{n-1}}+a_0\frac1{x^n}\right)$$ Now for large $x$ the quantity $a_{n-1}\frac1x+a_{n-2}\frac1{x^2}+\ldots+a_1\frac1{x^{n-1}}+a_0\frac1{x^n}$ is smaller in absolute value than $a_n$ (is close to $0$), therefore $a_n+a_{n-1}\frac1x+a_{n-2}\frac1{x^2}+\ldots+a_1\frac1{x^{n-1}}+a_0\frac1{x^n}$ is positive. To prove that, use that $$a_n>0\text{ and }\lim_{x\to\infty}a_n+a_{n-1}\frac1x+a_{n-2}\frac1{x^2}+\ldots+a_1\frac1{x^{n-1}}+a_0\frac1{x^n} =a_n$$ and this answer.