Let $X \subset$ $\mathbb{P} ^{n}$ be a hypersurface of degree $d$. Show that every line intersects $X$ in atmost $d$ points, unless the line is contained in $X$. Infact precisely $d$ points if counted with multiplicity.I have written a proof for this, can someone check whether this is correct? If not, please point out the step which have uncertainity.
Let $L$ be a line in $\mathbb{P} ^ {n}$. So, $L= \mathcal{Z } (f_0 (X_0,...,X_n ), ..., f_{n-1} (X_,...,X_n))$, where $f_i$ are homogeneous linear equations for $0\leq i\leq {n-1}$. They are linearly independent, so we can extend $\{ f_i (X_0,...,X_n ): 0\leq i\leq {n-1}\}$ to a basis of $k[X_0,...,X_n]$. With respect to new basis
$$X \cap L = \mathcal{Z}(f(f_0 ,...,f_n),f_0,...,f_{n-1})= \mathcal{Z}(f(0,...,0,f_n))$$ and since $f_n$ is linear, so degree of $f$ is $d$. Hence $f(0,...,0,f_n)$ can have atmost $d$ solutions.