MathDB

20

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(1)

2013 HMMT Guts #20: Polynomial Transformation of Roots

Source:

3/26/2013
The polynomial f(x)=x3−3x2−4x+4f(x)=x^3-3x^2-4x+4 has three real roots r1r_1, r2r_2, and r3r_3. Let g(x)=x3+ax2+bx+cg(x)=x^3+ax^2+bx+c be the polynomial which has roots s1s_1, s2s_2, and s3s_3, where s1=r1+r2z+r3z2s_1=r_1+r_2z+r_3z^2, s2=r1z+r2z2+r3s_2=r_1z+r_2z^2+r_3, s3=r1z2+r2+r3zs_3=r_1z^2+r_2+r_3z, and z=−1+i32z=\frac{-1+i\sqrt3}2. Find the real part of the sum of the coefficients of g(x)g(x).
HMMTalgebrapolynomialcomplex numbers