Problem
ALG-B1-M04-P009 A Hidden Zero Polynomial
#9
★★★★☆ Level 4 of 5
A polynomial \(P(x)\) of degree at most \(4\) equals \(0\) at five distinct integer values of \(x\). Prove that \(P\) is identically zero.
A non-zero polynomial of degree \(4\) cannot have more than four roots.
If \(P\) were non-zero, it would have at most \(4\) roots. But it has at least \(5\) roots. Contradiction. Hence \(P\equiv0\).
A mini-version of the interpolation principle.