Answer
(a) A polynomial of degree 3 has at most three real roots.
(b) A polynomial of degree $n$ has at most $n$ real roots.
Work Step by Step
(a) Let $f(x)$ be a polynomial of degree 3.
Then $f'(x)$ is a polynomial of degree 2.
Then there are at most 2 points $x_1$ and $x_2$ such that $f'(x) = 0$.
Since $f(x)$ is a polynomial, $f(x)$ is continuous and differentiable for all $x$.
Let's assume that the polynomial has at least four real roots $a_1$, $a_2$, $a_3$, and $a_4$ where $a_1 \lt a_2 \lt a_3 \lt a_4$. Then $f(a_1) = f(a_2) = f(a_3) = f(a_4) = 0$. According to Rolle's Theorem, there is a number $c_1$ in the interval $(a_1,a_2)$ such that $f'(c_1) = 0$, there is a number $c_2$ in the interval $(a_2,a_3)$ such that $f'(c_2) = 0$, and there is a number $c_3$ in the interval $(a_3,a_4)$ such that $f'(c_3) = 0$.
However, this contradicts the fact that $f'(x) = 0$ at two points at most.
Therefore, the polynomial has at most three real roots.
(b) Let $f(x)$ be a polynomial of degree $n$.
Then $f'(x)$ is a polynomial of degree $n-1$.
Then there are at most $n-1$ points $x_1$, $x_2$,...,$x_{n-1}$ such that $f'(x) = 0$.
Since $f(x)$ is a polynomial, $f(x)$ is continuous and differentiable for all $x$.
Let's assume that the polynomial has at least $n+1$ real roots $a_1$, $a_2$, $a_3$,...,$a_{n+1}$ where $a_1 \lt a_2 \lt a_3 \lt ...\lt a_{n+1}$. Then $f(a_1) = f(a_2) = f(a_3) =...= f(a_{n+1}) = 0$. According to Rolle's Theorem, there is a number $c_1$ in the interval $(a_1,a_2)$ such that $f'(c_1) = 0$, there is a number $c_2$ in the interval $(a_2,a_3)$ such that $f'(c_2) = 0$, ..., and there is a number $c_n$ in the interval $(a_n,a_{n+1})$ such that $f'(c_n) = 0$.
However, this contradicts the fact that $f'(x) = 0$ at $n-1$ points at most.
Therefore, the polynomial has at most $n$ real roots.
