Question 97
For every positive integer $n \geq 2$ the affirmation $P_{n}^{\prime}$: "If the derivative $P^{\prime}(X)$ of a polynomial $P(X)$ of degree $n$ with real coefficients has $n-1$ real distinct roots, then there exists a real constant $C$ such that the equation $P(x) = C$ has $n$ real distinct solutions" is considered. Prove that $P_{4}$ is true. Is the affirmation $P_{5}$ true? Prove the answer.