A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots

At degree $0$ the polynomial is a nonzero constant and has no root.

Assume the claim at degree $n$.

If a degree-$n+1$ polynomial has no root, its root count is already zero. Otherwise choose a root $a$ and write its actual coefficient list as $(c_0,\dots ,c_{n+1})$, with $c_{n+1}\mathrm{\ne }0$. Use the explicit quotient construction in A real polynomial vanishing at $a$ is divisible by $x-a$: its coefficient at $X^d$ is ${\sum }_{k=d+1}^{n+1}{c}_k{a}^{k-1-d}$ for $0\le d\le n$. The coefficient at $X^n$ is $c_{n+1}\mathrm{\ne }0$, and there are no higher terms, so this particular formal quotient $q$ has degree exactly $n$. For any other root $b\mathrm{\ne }a$, $0=p(b)=(b-a)q(b)$ and division by the nonzero real $b-a$ gives $q(b)=0$.

In the root case, the induction hypothesis in step 1.2 applied to the degree-$n$ quotient from step 1.3 gives at most $n$ distinct roots other than $a$, hence at most $n+1$ roots in all. The no-root case already satisfies the same bound. Induction proves the assertion at every degree.