Question 24
Let $p(x)$ be a polynomial of degree exactly $3$, with real coefficients. For each real number $a$, let $q_{a}(x)$ be the unique polynomial of degree $2$ or less such that $p(x)-q_{a}(x)$ is divisible by $(x-a)^{3}$. Prove that for $a \neq b$, the graphs of $q_{a}(x)$ and $q_{b}(x)$ do not intersect. Can you find a generalization to polynomials $p(x)$ of odd degree?