= Polynomial division preservation of exponential type
Let $F$ be an <entire function> of one variable satisfying $|F(\zeta)|\leq C(1+|\zeta|)^Ne^{R|\operatorname{Im}\zeta|}$, and let $P$ be a nonzero <polynomial>. If $F/P$ is entire, it satisfies a bound of the same exponential type, with a possibly different polynomial power. Indeed, outside a sufficiently large disk, $|P(\zeta)|\geq c(1+|\zeta|)^{\deg P}$; inside the disk the extended quotient is bounded. Combining these estimates proves the claim. The same reasoning works with the exact <support function> of a compact interval. Zeros of $F$ must cancel every root of $P$ with its full multiplicity.
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