Unique Monic Polynomials

Algebra Level 5

How many monic polynomials with real coefficients of degree \(<1000\) have the property that \(f(x^2+x)=(f(x))^2+f(x)\) for all \(x\)?

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.

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