proving that a polynomial has at least two roots

Question

I have the following question in real analysis course:
let $p(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$. Prove that if $x_{0}$ is a root of P, meaning $p(x_{0})=0$ and $p'(x_{0})\neq0$, then p has at least two roots.
approach: I tried to make several things, like first of looking at $p(-x_{0})$ and denote $p(-x_{0})=\frac{m}{2}$ and so $\displaystyle\frac{m}{2}=x_{0}^{4}-a_{3}x_{0}^{3}+a_{2}x_{0}^{2}-a_{1}x_{0}+a_{0}$ and then we can use the fact that $p(x_{0})=0$ and therefore $p(x_{0})-p(-x_{0})=-\frac{m}{2}$. It doesn't get me much far tho.
I assume it has to do something with the derivative because they mentioned it. any clues how to help?
Thank you all in advance

Answer 1

It is interesting that you listed infinitesimals as one of the tags for this question, even though you did not mention them in the body of the question. Infinitesimals are indeed relevant here, as follows. Since the derivative at $x_0$ is nonzero, there are two cases: (a) $p'(x_0)<0$, and (b) $p'(x_0)>0$. Let's consider for example case (a). Since the derivative is negative, for all infinitesimal $\epsilon>0$ one will have $p(x_0+\epsilon)<0$. On the other hand, since the polynomial is monic, for sufficiently large $N$ one will have $p(x)>0$ when $x> N$. By the intermediate value theorem, the polynomial must have another zero between $x_0+\epsilon$ and $N$.

Answer 2

Let p(x) be a polynomial of exact even degree n >= 2. Assume that the highest power has a coefficient of 1, otherwise we divide by the highest coefficient. Now if we calculate the limit if p(x) for x -> +infinity and x -> -infinity, both are +infinity, so we have two different points with p(x) > 0.

Your condition shows that you have t with p(t) = 0, and p(t') < 0 for either some t' < t or t' > t. Either way the polynomial has both positive and negative values in an interval that doesn't include t, and because polynomials are continuous there must be a zero at some point other than t.

You actually don't need polynomials and converging towards infinity, all you need is that p(x) is continuous everywhere, and there are arbitrary large and arbitrary small x both with p(x) > 0, or both with p(x) < 0.

For odd degrees it doesn't work; for example for $p(x) = x$, or $p(x) = x^3 + x$, or $p(x) = x^{999} + x$.