Zeroes of a $n$-variable boolean function of degree $d$.

Question

Let $f$ be a non-constant boolean function of degree $d$ in variables $x_1,x_2,\dots,x_n$ where $d < n$. How do we prove that there exists a non-zero vector with at most $d+1$ many $1$ such that $f(v)=0$??

1. I tried using induction on the degree of the polynomial. I tried expressing $f = x_n(f_1) + f_2$ but failed at attaining a vector that annihilates both $f_1$ and $f_2$. Also, I have no idea to construct a vector with $f_1(v) = f_2(v) = 1$.
2. Other approach I tried was to construct a matrix $A_{n\times n}$ of rank at most $d$ such that its nullvector is a zero of the polynomial $f$.

The proof is expected using linear algebra methods.