  The Waring Problem over polynomial rings asks for how to decompose an
homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of
linear forms.
  First, we give a constructive method to obtain a real Waring decomposition of
any given real binary form with length at most its degree. Secondly, we adapt
the Sylvester's Algorithm to the real case in order to determine a Waring
decomposition with minimal length and then we establish its real rank. We use
bezoutian matrices to achieve a minimal decomposition.
  We consider all real binary forms of a given degree and we decompose this
space as a finite union of semialgebraic sets according to their real rank.
Some examples are included.
