Is there a way to determine if a polynomial has only real roots?

[Klaus_Hoffmann]

given a Polynomial or a trigonometric Polynomial

$$K(z)= \sum_{n=0}^{N}a_{n}x^{n}$$ and

$$H(x)= \sum_{n=0}^{N}b_{n}e^{inx}$$

is there a criterion to decide or to see if K(z) or H(x) have ONLY real roots

 

[mathman]

For the ordinary polynomial there is a procedure involving generating a Sturm sequence (gets messy for large N) which can be used to determine the number of real roots greater than a given value of x. To get what you want, use a sufficiently large negative x, i.e. look at the highest order term in each of the polynomials in the sequence (there will be N+1).

 

[tacman]

Yes, there are criteria that can be used to determine if a polynomial has only real roots. One such criterion is the Descartes' rule of signs. This rule states that the number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial, and the number of negative real roots is equal to the number of sign changes in the coefficients of the polynomial with the signs reversed. This can be used to determine if a polynomial has only real roots by checking the number of sign changes and comparing it to the degree of the polynomial.

For trigonometric polynomials, there is a similar criterion known as the Fourier-Descartes theorem. This theorem states that the number of real roots of a trigonometric polynomial is equal to the number of sign changes in the coefficients of the polynomial, with the exception of the constant term. This can also be used to determine if a trigonometric polynomial has only real roots.

In addition to these criteria, there are other methods such as using the quadratic formula or graphing the polynomial to visually see if it has any complex roots. However, these methods may not always be accurate and can be time-consuming. Therefore, the Descartes' rule of signs and the Fourier-Descartes theorem are reliable and efficient ways to determine if a polynomial or a trigonometric polynomial has only real roots.