# Polynomials and squares

Algebra Level 4

Consider all monic quadratic polynomials $$f(x)$$ with real coefficients such that $g(x) = (f(x))^2-f(x^2)$ is also monic, and has exactly three non-zero coefficients.

The sum of $$f(0)$$ for all such quadratic polynomials can be written as $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers. What is the value of $$m+n$$?

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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