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Find all polynomials $P(X)$ of fourth degree with real coefficients that verify

Source: Moldova TST 1996

August 8, 2023
algebrapolynomial

Problem Statement

Find all polynomials P(X)P(X) of fourth degree with real coefficients that verify the properties: a) P(−x)=P(x),∀x∈R;P(-x)=P(x), \forall x\in\mathbb{R}; b) P(x)≥0,∀x∈R;P(x)\geq0, \forall x\in\mathbb{R}; c) P(0)=1;P(0)=1; d) P(X)P(X) has exactly two local minimums x1x_1 and x2x_2 such that ∣x1−x2∣=2.|x_1-x_2|=2.
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