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 r1​r2​=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 r3​r4​=r1​r2​​=13−i and r1​+r2​=r3​+r4​​=3−4i. The polynomial is therefore