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Question

$P(x)$ is a fourth-degree polynomial with real coefficients. Given that, $$ \forall x \in \mathbb{R}, P(x) \ge x $$ and that, $$ P(1)=1, P(2)=4, P(3)=3 $$ What is $P(4)$?

Answer

There is a beautiful solution with the help of the following graph:

enter image description here

We can reason that $P(x)+x$ must have double roots at $x=1$ and $x=3$, so $P(x)=k(x-1)^2 (x-3)^2 + x$, and we find $k$, etc.

In what other ways one can reason without using a graph?

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    I think you mean "We can reason that $P(x) - x$ must have ..." – Arthur Jan 29 '24 at 10:42
  • The graph isn't actually used for reasoning, more to illustrate the reasoning. I don't think there's going to be a neater way to do it than deducing that $P(x)-x$ is quartic and has two double roots, as this is the most direct use of the given information. (Had you been told, say, that $P$ had integer coefficients, there would be other approaches.) – Chris Lewis Jan 29 '24 at 11:53

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