polynomial of order n whose coefiicients  belong to Z and alternates signs.

Do you mean all roots are real? 

yes

There is a criterion due to J. Sylvester: The number of real roots of a polynomial f(x) equals the signature of its Bezoutiant. In particular a real polynomial has all its roots real and distinct if and only if the Bezoutiant is positive definite. See for example: C. Procesi, Lie Groups, Springer, 2005, page 25.

Condition: If a polynomial is a characteristic polynomial of a symmetric matrix then all its roots are real.

See Thm. 4.1 in Dubickas - Sha, Counting and testing dominant polynomials,

Experimental Mathematics, 24:312-325 (2015).

Also, roots of orthogonal polynomials (e.g. Chebyshev polynomials) are real.

You can find the answer (also to many other questions) in

Hermite's Method of Separation ...  

Uteshev A.Y and Shulyak S.G.

Linear Algebra and its Applications 177:  49-88, (1992) 

Best regards

Dr. Dieter Kaesbauer

Look this link; perhaps it could help you somthing.

Although not exactly what you ask, you might be interested in the algorithmic method attached for finding integer roots in polynomials with coefficients in Z (in Spanish):

ALGORITMO DE DESCARTE DE RAÍCES ENTERAS DE POLINOMIOS - Jesús Álvarez Lobo. RG.

http://ac.els-cdn.com/002240499090108T/1-s2.0-002240499090108T-main.pdf?_tid=9fc4db62-eac2-11e5-8806-00000aacb362&acdnat=1458055913_549a091d9a19ce1b0cee55f045858ae0 

Article ALGORITMO DE DESCARTE DE RAÍCES ENTERAS DE POLINOMIOS

This should completely answer your question. You may also want to consult Strum theorem.

http://mathoverflow.net/questions/20946/criteria-to-determine-whether-a-real-coefficient-polynomial-has-real-root

This should completely answer your question. You may also want to consult Strum theorem.

http://mathoverflow.net/questions/20946/criteria-to-determine-whether-a-real-coefficient-polynomial-has-real-root