= Solution
On $\mathbb C^{n+1}\setminus\{0\}$, let $\Omega=dz_0\wedge\cdots\wedge dz_n$ and let $E=\sum_i z_i\partial_{z_i}$ be the <Euler vector field>. The numerator is its <interior product of a differential form>, so
$$
\alpha=\frac{\iota_E\Omega}{f}.
$$
Both numerator and denominator have homogeneous scaling degree $n+1$. Thus this meromorphic $n$-form is invariant under constant nonzero scalings, and $\iota_E\alpha=0$ makes it horizontal for the quotient to <Complex projective space>.
To check descent using the hint, take a local holomorphic section $Z$ of the quotient and another section $Z'=\lambda Z$, where $\lambda$ is a nowhere-zero <holomorphic function>. In the evaluation
$$
(Z')^*(\iota_E\Omega)(v_1,\ldots,v_n)
=\Omega\bigl(\lambda Z,\lambda\,dZ(v_1)+d\lambda(v_1)Z,\ldots,\lambda\,dZ(v_n)+d\lambda(v_n)Z\bigr),
$$
every term involving $d\lambda$ has two collinear arguments and vanishes. The remaining term is $\lambda^{n+1}Z^*(\iota_E\Omega)$, while $f(\lambda Z)=\lambda^{n+1}f(Z)$. Hence $Z'^*\alpha=Z^*\alpha$. The forms therefore glue to a meromorphic section of the <canonical bundle> of <Complex projective space>.
On the affine chart $z_j\ne0$, use the section $z_j=1$ and list the remaining coordinates as $t_1,\ldots,t_n$ in increasing original-index order. If $F$ is the resulting polynomial, then
$$
\alpha=(-1)^j\frac{dt_1\wedge\cdots\wedge dt_n}{F}.
$$
At a point of $Y$, some derivative $F_{t_r}$ is nonzero. Otherwise all the homogeneous derivatives except possibly $f_{z_j}$ would vanish; the <Euler homogeneous function theorem> gives $\sum_i z_if_{z_i}=(n+1)f=0$ there, forcing $f_{z_j}=0$ too. This contradicts the permitted smoothness criterion. The <holomorphic inverse function theorem> now makes $F$ a transverse coordinate, so the pole is simple. On a neighbourhood with $F_{t_r}\ne0$, rearranging the <wedge product of differential forms> gives
$$
\alpha=\frac{dF}{F}\wedge
\frac{(-1)^{j+r-1}}{F_{t_r}}\,dt_1\wedge\cdots\wedge\widehat{dt_r}\wedge\cdots\wedge dt_n.
$$
By part (a), its <Poincaré residue> is consequently
$$
\boxed{\operatorname{Res}_Y(\alpha)=
\left.\frac{(-1)^{j+r-1}}{F_{t_r}}\,dt_1\wedge\cdots\wedge\widehat{dt_r}\wedge\cdots\wedge dt_n\right|_Y.}
$$
The remaining $t$ coordinates form a <holomorphic coordinate> system on $Y$, and $F_{t_r}$ is nonzero in this neighbourhood. The displayed top form is therefore nowhere zero. These neighbourhoods cover $Y$ and part (a) guarantees agreement on their overlaps. \b[The residue is a nowhere-vanishing global holomorphic section]:
$$
\boxed{K_Y\cong\mathcal O_Y.}
$$
This is the explicit <residue trivialization for a degree n+1 projective hypersurface>, consistent with the <adjunction formula> $K_Y\cong\mathcal O_Y((n+1)-n-1)$, but obtained here directly from the indicated form.
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