* problem*: Consider the equation \({\displaystyle{x^3 + Gx + H = 0}} \), where G and H are complex numbers. Suppose that this equation has a pair of complex conjugate roots. Show that both G and H are real.

* solution*: Let three roots of the equation \({\displaystyle{x^3 + Gx + H = 0}} \)

are \({\displaystyle{\alpha, \beta, \gamma}} \) [ Let \({\displaystyle{\alpha, \beta}} \) are complex conjugates]

Now \({\displaystyle{\alpha \beta \gamma}} \) = – H … (i)

\({\displaystyle{\alpha + \beta + \gamma}} \) = 0 … (ii)

\({\displaystyle{\alpha \beta + \beta \gamma + \gamma \alpha}} \) = G … (iii)

From (ii) we get

\({\displaystyle{\alpha + \beta + \gamma}} \) = 0 [ \({\displaystyle{\alpha, \beta}} \) are complex conjugates so they are real]

\({\Rightarrow} \) \({\gamma} \) = real

Now as \({\gamma} \) = real

\({\displaystyle{\beta \gamma + \gamma \alpha}} \) = \({\displaystyle{\gamma (\beta + \alpha)}} \)

= \({real \times real} \)

= \({real} \) … (iv)

\({\alpha, \beta} \) are complex conjugates so \({\alpha \beta = real} \) … (v)

From (iv) & (v) we get \({\displaystyle{\alpha \beta + \beta \gamma + \gamma \alpha}} \) = \({real + real = real} \)

\({\Rightarrow} \) G = real [from (iii)]

Now \({\alpha, \beta} \) is real and \({\alpha, \beta} \) is real

so \({\displaystyle{\alpha \beta \gamma}} \) = real

\({\Rightarrow} \) H = real.