Problem: Let $V$ be the real vector space of all real polynomials in one variable, and let $P : V \to \mathbb{R} $ be a linear map. Suppose that for all $f, g \in V$ with $P(fg) = 0$ we have $P(f) = 0$ or $P(g) = 0$. Prove that there exist real numbers $x_0$ and $c$ such that $P(f) = c f(x_0)$ for all $f\in V$.
Solution: I encourage you to attempt to solve the problem today. The solution will be provided tomorrow. This will give you the opportunity to test your understanding of the problem and to improve your skills in solving similar problems in the future.