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incredible polynomial, 10/10

Source: Moldova TST 2019

March 10, 2019
polynomialalgebra

Problem Statement

Find all polynomials P(X)P(X) with real coefficients such that if real numbers x,yx,y and zz satisfy x+y+z=0,x+y+z=0, then the points (x,P(x)),(y,P(y)),(z,P(z))\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right) are all colinear.
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