Can Two Different Polynomials Agree on an open interval?

Question

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)?

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TBN: This question is partially answered in :Overlapping Polynomials however I need to know if it is true for all polynomials of any degree.

Answer 1

Only if $P_1=P_2$ and that holds for any degrees

Answer 2

Yes. Assume that $P_1$ and $P_2$ are equal on some open interval and let the degrees of $P_1$ and $P_2$ be at most $n$. Then $P_1 - P_2$ is a polynomial of degree at most $n$ that has at least $n + 1$ distinct roots [since every number in the interval is a root]. But a nonzero polynomial of degree at most $n$ has at most $n$ roots.