Generalization of real roots for a polynomial of $n$ degree.

Question

Is there any specific condition on the coefficients of a polynomial of $n$ degree, so that all roots are real ?

I know that there is a condition on quadratic polynomials : $p(x) = ax^2+bx+c$, for both the roots to be real, $b^2-4ac\ge0$.

Answer 1

Yes, there are some conditions. Such as, if

$$P(x) = \sum_{i=0}^n a_i x^i,$$ then

If $a_i - 4 a_{i-1} a_{i+1} \geq 0,$ for all $i=1\dotsc, n-1,$ then the roots of $P(x) are all real and distinct.

For proof, see:

Kurtz, David C., A sufficient condition for all the roots of a polynomial to be real, Am. Math. Mon. 99, No. 3, 259-263 (1992). ZBL0761.26011.