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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 ${displaystyle{x^3 + Gx + H = 0}}$, 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.