- #1
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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.
$$
-\dfrac{a_{n-1}}{n} \pm \dfrac{n-1}{n}\sqrt{a_{n-1}^2-\dfrac{2n}{n-1}a_{n-2}}\,.
$$
Hint: Use the inequality of Cauchy-Schwarz.