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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