An algebra problem by Thomas Jacob

Consider the function

$\begin{aligned} f(x) & = a_0 + a_1x + a_2x^2 + a_3x^3 + ... + a_nx^n \\ f \left(e^{\frac {2\pi}3 i}\right) & = a_0 + a_1e^{\frac {2\pi}3 i} + a_2e^{\frac {4 \pi} 3 i} + a_3e^{2\pi i} + ... + a_ne^{\frac {2n \pi} 3 i} \\ & = a_0 + a_1 \left(\cos \frac {2\pi}3 + i \sin \frac {2\pi}3 \right) + a_2 \left(\cos \frac {4\pi} 3 + i \sin \frac {4\pi} 3 \right) + a_3\left(\cos 2\pi + i \sin 2\pi \right) + ... + a_n \left(\cos \frac {2n\pi} 3 + i \sin \frac {2n\pi} 3 \right) \\ \Re \left \{f \left(e^{\frac {2\pi}3 i}\right) \right \} & = a_0 + a_1 \cos \frac {2\pi}3 + a_2 \cos \frac {4\pi} 3 + a_3 \cos 2\pi + ... + a_n \cos \frac {2n\pi} 3 \\ & = a_0 - \frac {a_1}2 - \frac {a_2}2 + a_3 - ... + a_n \cos \frac {2n\pi} 3 \\ \implies A & = \Re \left \{f \left(e^{\frac {2\pi}3 i}\right) \right \} \\ & = \Re \left \{\left(e^{\frac {3916 \pi} 3 i} + e^{\frac {3914 \pi} 3 i} + 2\right)^{1959} \right \} \\ & = \Re \left \{\left(e^{\frac {4\pi} 3 i} + e^{\frac {2\pi} 3 i} + 2\right)^{1959} \right \} \\ & = \Re \left \{\left(-\frac 12 -\frac {\sqrt 3}2i -\frac 12 + \frac {\sqrt 3}2i + 2\right)^{1959} \right \} \\ & = \Re \left \{\left( 1 \right)^{1959} \right \} \\ & = 1 \end{aligned}$

$\implies 2A+3 = \boxed{5}$

2A + 3 = f(ω) + f(ω^2) +3