(open) Boundaries on the roots of splitting real polynomials

[fresh_42]

Suppose all roots of the polynomial $x^n+a_{n−1}x^{n−1}+\cdots+a_0$ are real. Then the roots are contained in the interval with the endpoints
$$
-\dfrac{a_{n-1}}{n} \pm \dfrac{n-1}{n}\sqrt{a_{n-1}^2-\dfrac{2n}{n-1}a_{n-2}}\,.
$$
Hint: Use the inequality of Cauchy-Schwarz.

 

[anuttarasammyak]

For n=2 the formula is familiar quadric equation solution.

 

[fresh_42]

For n=2 the formula is familiar quadric equation solution.

... and this observation is one hint to the proof!