Multiplicities of poles of a divisor of a rational function w.r.t. an elliptic curve

I am reading Sec 5.8.2 in the textbook Introduction to Mathematical Cryptology (Hoffstein, Pipher and Silverman), a precursor to introducing the structure of Weil pairing. It first defines a rational function in one variable, $f(x)$ its corresponding zeroes and poles and uses that to define the $div(f(x))$. They move on to elliptic curves. They define an $E: y^2 = x^3+ax+b$ and consider a rational function in two variables $f(x,y)$ and define poles and zeros of $f$ on $E$ as the points of $E$ where the denominator and numerator of $f$ disappear, respectively. They then consider the example (Example 5.35) where $x^3+ax+b = (x-\alpha_1)(x-\alpha_2)(x-\alpha_3)$. They define $P_1 = (\alpha_1,0)$, $P_2 = (\alpha_2,0)$, $P_3 = (\alpha_3,0)$ amd note that they are of order $2$. They look at the function $y$ and say that it disappears at $P_1, P_2,P_3$, which means they are zeros. They then go on to define the divisor of $y$ as $div(y) = [P_1]+[P_2]+[P_3] - 3[\mathcal{O}]$.

I am not able to understand how they concluded $\mathcal{O}$ is a pole of $y$ and it has multiplicity $3$.