This is a Test of Mathematics Solution Subjective 73 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Problem

Consider the equation $ {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 $ {x^3 + Gx + H = 0} $ are

$ {\alpha, \beta, \gamma} $   [ Let $ {\alpha, \beta} $ are complex conjugates] .Now ${\alpha \beta \gamma} $ = - H ... (i)

$ {\alpha + \beta +\gamma} $ = 0 ... (ii)

$ {\alpha \beta + \beta \gamma + \gamma \alpha} $ = G ... (iii)

From (ii) we get $ {\alpha + \beta + \gamma}$

= 0 [ $ {\alpha, \beta} $ are complex conjugates so they are real]

${\Rightarrow} $ $ {\gamma} $ = real Now as $ {\gamma} $ = real

$ {\beta \gamma + \gamma \alpha} $ = $ {\gamma (\beta + \alpha)} $ =$ {real \times real} $ = $ {real} $ ... (iv) $ {\alpha, \beta} $ are complex conjugates so $ {\alpha \beta = real} $ ... (v) From (iv) & (v) we get

$ {\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 ${\alpha \beta \gamma} $ = real    $ {\Rightarrow}$ H = real.