We generalize the classical Vieta formulas that express the coefficients of a polynomial in terms of all the roots. In particular, we focus on the case when just some of the roots are known. Our formulas are established by exploiting    some properties of the complete homogeneous symmetric polynomials. 
Further, we provide  new identities for these polynomials when the variables are in a geometric progression and give an application to cyclotomic polynomials. This is joint work with  A. Echezabal and M. Laporta