Test of Mathematics at the 10+2 Level 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.