Not all the roots are real iff a^2_1<a_2

[lfdahl]

Given the equation
$$x^n + a_1x^{n-1}+a_2x^{n-2}+…+a_n = 0$$
- with real coefficients, and $a_1^2 < a_2$.

Show that not all the roots are real.

 

[Opalg]

Given the equation
$$x^n + a_1x^{n-1}+a_2x^{n-2}+…+a_n = 0$$
- with real coefficients, and $a_1^2 < a_2$.

Show that not all the roots are real.

[sp]If the roots are $x_1,x_2,\ldots,x_n$ then by one of Newton's identities $x_1^2 + x_2^2 + \ldots + x_n^2 = a_1^2 - a_2$. If all the roots are real then the sum of their squares would have to be positive. But if $a_1^2 < a_2$ then that sum is negative. So not all the roots can be real.

[/sp]

 

[lfdahl]

[sp]If the roots are $x_1,x_2,\ldots,x_n$ then by one of Newton's identities $x_1^2 + x_2^2 + \ldots + x_n^2 = a_1^2 - a_2$. If all the roots are real then the sum of their squares would have to be positive. But if $a_1^2 < a_2$ then that sum is negative. So not all the roots can be real.

[/sp]

Thankyou, Opalg, for your participation. Your solution is - of course - correct.(Cool)