For a smooth hypersurface $X_d^n \subset \mathbb{P}^{n+1}$ of degree $d$ and dimension $n$, the topological Euler characteristic is the polynomial in $d$

a consequence of the Hirzebruch–Riemann–Roch computation of the Chern classes of a hypersurface. That geometric input is taken from the literature; this concept is about the polynomial itself. Multiplying by $d$ removes the division: $chiPoly n d = (1 - d)^(n+2) - 1 + (n+2) d$.

The statement is that this polynomial equals the tail of the binomial expansion of $(1-d)^{n+2}$ from the quadratic term on, $\sum_{k=0}^{n} \binom{n+2}{k+2} (-d)^{k+2}$: the constant and linear terms of the binomial cancel against $-1$ and $(n+2)d$, which is why $\chi$ is a genuine polynomial with no constant or linear term. The familiar cases are $d^3 - 4d^2 + 6d$ for surfaces and $10d - 10d^2 + 5d^3 - d^4$ for threefolds.