Problem:
For certain real values of a,b,c, and d, the equation x4+ax3+bx2+cx+d=0 has four non-real roots. The product of two of these roots is 13+i and the sum of the other two roots is 3+4i, where i=−1. Find b.
Solution:
Let the roots be r1,r2,r3,r4, where r1r2=13+i and r3+r4=3+4i. Because the polynomial has real coefficients and none of the roots is real, the roots occur in conjugate pairs, say r3=r1 and r4=r2. It follows that r3r4=r1r2=13−i and r1+r2=r3+r4=3−4i. The polynomial is therefore