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February 15, 2021

Test of Mathematics Solution Subjective 70 - Equal Roots

Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 70 (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

Suppose that all roots of the polynomial equation
$ {\displaystyle{x^4 - 4x^3 + ax^2 +bx + 1}} $ = 0 are positive real numbers.
Show that all the roots of the polynomial are equal.


Solution


$ {\displaystyle{x^4 - 4x^3 + ax^2 +bx + 1}} $ = 0
If the roots are $ {\displaystyle{\alpha}} $, $ {\displaystyle{\beta}} $, $ {\displaystyle{\gamma}} $ and $ {\displaystyle{\lambda}} $ .
then $ {\displaystyle{\alpha}} $, $ {\displaystyle{\beta}} $, $ {\displaystyle{\gamma}} $ and $ {\displaystyle{\lambda}} $ = 1
& $ {\displaystyle{\alpha}} $ + $ {\displaystyle{\beta}} $ + $ {\displaystyle{\gamma}} $ + $ {\displaystyle{\lambda}} $ = 4.
Now all of $ {\displaystyle{\alpha}} $, $ {\displaystyle{\beta}} $, $ {\displaystyle{\gamma}} $ and $ {\displaystyle{\lambda}} $ are positive so AM-GM inequality is applicable.
$ {\displaystyle{\frac{\alpha + \beta + \gamma + \lambda}{4}}}{\ge}$ $ {(\alpha\beta\lambda)^{\frac{1}{4}}}$
$ {\Rightarrow} $ $ {\frac{4}{4}} {\ge}$ $ {1^{\frac{1}{4}}}$
$ {\Rightarrow} $ 1 $ {\ge} $ 1
Now we know equality in AM-GM occours if all the numbers are equal.So $ {\displaystyle{\alpha}} $, $ {\displaystyle{\beta}} $, $ {\displaystyle{\gamma}} $ and $ {\displaystyle{\lambda}} $ are all equal.

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