Family Functions Related to Phi

Algebra Level 3

Let \(f_n(x)\) be a family of quadratics with integer coefficients in the form \(ax^2+bx+c\) such that at least one of its roots is \(\phi\), which is \(\frac{1+\sqrt{5}}{2}\). There exists a quadratic in the family \(f_n(x)\) such that the sum \(a+b+c=-1\). The name of this particular function is \(f_{\star}(x)\). Find \(f_{\star}(16)\).

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