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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.


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

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.

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.


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

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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