Prime ideals in $A[x_1, \ldots,x_n]$

Question

Let $A$ be a commutative, Noetherian ring and let us define a monomial ordering, $\prec$ on $A[x_1, \ldots,x_n]$. My doubt is regarding the maximal chain of prime ideals in $A[x_1, \ldots,x_n]$. When we look at any one of the generators, $f$ of a prime ideal $P_i$ in $A[x_1, \ldots, x_n]$ such that $f\notin A$, is it enough to consider $f$ such that $\mathrm{lc}(f) = 1$? Will the coefficients increase the length of a chain of prime ideals?