Mathematics

Mathematics

Problem

Let f(x)=x4+ax3+bx2+cf(x) = x^4 + ax^3 + bx^2 + c be a polynomial with real coefficients such that f(1)=−9f(1) = -9. Suppose that i3i\sqrt{3} is a root of the equation 4x3+3ax2+2bx=04x^3 + 3ax^2 + 2bx = 0, where i=−1i = \sqrt{-1}. If α1,α2,α3\alpha_1, \alpha_2, \alpha_3, and α4\alpha_4 are all the roots of the equation f(x)=0f(x) = 0, then ∣α1∣2+∣α2∣2+∣α3∣2+∣α4∣2|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2 is equal to ________.

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