real analysis - Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$.

Prove that the reciprocal of a polynomial function $f(x)$ is uniformly continuous on $R$.

(It is provided that the reciprocal of the function exists. In other words, $f(x)$ is never zero for any value of $x$.)

I go by the way:

Let $g(x) = \frac{1}{f(x)}$, where $f(x) = a_0 + a_1x + a_2x^2 + . . . + a_nx^n.$

Now we have to show that $g(x)$ is uniformly contnuous on $R.$

Then

$|g(x)-g(x_o)| = |\frac{1}{f(x)} - \frac{1}{f(x_0)}|$
$=|x - x_0||\frac{a_1(x - x_0) + a_2(x^2 - x_0^2) + . . . + a_n(x^n - x_0^n)}{f(x)f(x_0)}|$

Then how to proceed??