# Meaning and/or source of polynomials residually of the form $x^n(x-1)$ in Gabber's characterization of Henselian pairs?

Lemma 09XI in the stacks project includes a characterization (#5) by Gabber of Henselian pairs $$(A,I)$$: first, $$I\leq \mathrm J(A)$$ is contained in the Jacobson radical and second, every monic $$f\in A[x]$$ which is residually of the form $$x^{\deg f-1}(x-1)\in \frac{A}{I}[x]$$ has a root in $$1+I$$.

The proof uses this condition #5 to prove an $$A$$-algebra morphism from an étale $$A$$-algebra to $$A/I$$ lifts to a retraction of the étale $$A$$-algebra. The actual source for a polynomial as above is lemma 0EM0, whose statement and proof are fairly involved and opaque to me.

Question. Let $$A$$ be a commutative ring and $$I\leq \mathrm J(A)$$ an ideal contained in the Jacobson radical. Where do monics $$f\in A[x]$$ which are residually of the form $$x^{\deg f-1}(x-1)\in \frac{A}{I}[x]$$ "come from" conceptually? Algebraic and geometric answers are both welcome.