INTRODUCING 5 - days-a-week problem solving session for Math Olympiad and ISI Entrance. Learn More

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

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.

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.

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

JOIN TRIAL