Complemented subspaces of homogeneous polynomials

Let ${\mathcal{P}}_K{(}^{n}E;F)$ (resp. ${\mathcal{P}}_w{(}^{n}E;F)$) the subspace of all $P\in \mathcal{P}{(}^{n}E;F)$ which are compact (resp. weakly continuous on bounded sets). We show that if ${\mathcal{P}}_K{(}^{n}E;F)$ contains an isomorphic copy of $c_0$, then ${\mathcal{P}}_K{(}^{n}E;F)$ is not complemented in $\mathcal{P}{(}^{n}E;F)$. Likewise we show that if ${\mathcal{P}}_w{(}^{n}E;F)$ contains an isomorphic copy of $c_0$, then ${\mathcal{P}}_w{(}^{n}E;F)$ is not complemented in $\mathcal{P}{(}^{n}E;F)$.