Roots of homogeneous polynomials in $\mathbb{P}^n$

Question

Let $k$ be an algebraically closed field and $d>0$.

Question: Let $a\in \mathbb{P}^n$. Does there exist a nonzero degree $d$ homogeneous polynomial $f\in k[x_0,\dots,x_n]$ such that $f(a)=0$?

For $d=1$ and $d=n$, I think this is true. Is it true for all $d>0$? If so, how can we find $f$?

Answer 1

Hint You've already said that you can produce such polynomial $f_1$ of degree $1$ with the desired property. If $d > 1$, then for any polynomial $f_{d - 1}$ homogeneous of degree $d - 1$...

Additional hint Show that the product $f := f_1 f_{d - 1}$ satisfies the given conditions.