Given that the complex number z 2 7i is a root to the equa

Given that the complex number z = -2 + 7i is a root to the equation:

z³ + 6 z² + 61 z + 106 = 0

find the real root to the equation.

Solution

Since z = -2 + 7i is a root to the equation and all the coefficients in the terms of the equation are real numbers, then z\' the complex conjugate of z is also a solution. Hence

z³ + 6 z² + 61 z + 106 = (z - (-2 + 7i))(z - (-2 - 7i)) q(z)

= (z² + 4z + 53) q(z)

q(z) = [ z³ + 6 z² + 61 z + 106 ] / [ z² + 4z + 53 ] = z + 2

Z + 2 is a factor of z³ + 6 z² + 61 z + 106 and therefore z = -2 is the real root of the given equation.