In the proof of the Nullstellensatz for projective varieties, I can't understand the following remark (which comes from Hulek, Elementary algebraic geometry, pag. 72)
" ... if $f=\sum f_i$ is a polynomial with homogeneous components $f_i$ then (using the fact that the field $k$ has infinitely many elements) $f(\lambda x_0,\ldots,\lambda x_n)=0$ for all $\lambda$ if and only if $f_i(x_0,\ldots,x_n)=0$ for all $i$...."
The only if part is trivial. I can't understand the if part. Can you help?
Hint: Can you see why the equality $f_i(\lambda x_0,\ldots,\lambda x_n)=\lambda^if_i(x_0,\ldots,x_n)$ holds for all $i$?