Birkhoff-Grothendieck theorem states that any holomorphic vector bundle over $\mathbb P^1$ splits as a direct sum $E = \bigoplus \mathcal O_{\mathbb P^1}(a_i)$. I am trying to work out an explicit example of this and failing so far.
Let $f_0, f_1, f_2$ be homogeneous polynomials in $(x_0:x_1)$ with no common roots. Then homomorphism $f: \mathcal O^{\oplus 3} \to \mathcal O(n) : (f_0, f_1, f_2)$ has constant rank $1$, therefore, $E = \mathrm{ker}(f)$ is vector sub-bundle of $\mathcal O^{\oplus 3}$.
I am interested in the splitting type of $E$. Compluting the Chern class, one can see that if $E \simeq \mathcal O(a) \oplus \mathcal O(b)$, then $a+b=-n$. My intuition suggests that $E \simeq \mathcal O \oplus \mathcal O(-n)$, but I have not been able to find desired isomorphism.