Test of Mathematics Solution Subjective 70 - Equal Roots

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.

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.