Weird Polynomial Relation

Algebra Level 2

Let \(P(x)\) a polynomial in \(x\) with real coefficients such that for all real numbers \(x, y, z\) satisfying \(xy+yz+zx=1,\) \[P(x)+P(y)+P(z)=P(x+y+z).\] Furthermore, \(P(0)=1\) and \(P(1)=4.\) What is the value of \(\sqrt{P(2017)}?\)

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