  A Banach space $E$ has the Grothendieck property if every (linear bounded)
operator from $E$ into $c_0$ is weakly compact. It is proved that, for an
integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly
compact if and only if the space ${\cal P}(^kE)$ of scalar valued polynomials
on $E$ is reflexive. This is equivalent to the symmetric $k$-fold projective
tensor product of $E$ (i.e., the predual of ${\cal P}(^kE)$) having the
Grothendieck property. The Grothendieck property of the projective tensor
product $E\widehat{\bigotimes}F$ is also characterized. Moreover, the
Grothendieck property of $E$ is described in terms of sequences of polynomials.
  Finally, it is shown that if every operator from $E$ into $c_0$ is completely
continuous, then so is every polynomial between these spaces.
