Answer to Question #103675 in Algebra for Sourav Mondal

Question

Use the Cauchy-Schwarz inequality to solve x³-25x²-4x+100=0, if we know
that all its roots are rational.

Answer 1

As per the given question,

$x^3-25x^2-4x+100=0,$

Now,

$x^3-4x-25x^2+100=0,$

$x(x^2-4)-25(x^2-4)=0,$

$(x^2-4)(x-25)=0,$

so,

$x^2=4$ or $x=25$,

$x=\pm2$ or $x=25$.