Answer to Question #114745 in Algebra for ANJU JAYACHANDRAN

Question

Use the Cauchy-Schwarz inequality to solve x^3-25x^2-4x+100=0, if we know
that all its roots are rational.

Answer 1

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

imagine the left side as factors

$(x-2)(x-25)(x+2)=0$

each bracket may be equal 0. thats we have next three equations:

$x-2=0\\ x-25=0\\ x+2=0$

This equations have three roots: $x=2$, $x=25$, $x=-2$

This equation cannot be solved using the Cauchy-Schwartz inequality, since we have equality, and in the Cauchy-Schwarz inequality two variables are used, and we have one.