Integer partitions express the different ways that a positive integer may be written as a sum of positive integers. Here we explore the analytic properties of a new polynomial f_λ(x) that we call the partition polynomial for the partition λ, with the aim to learn new properties of partitions. We prove a recursive formula for the derivatives of f_λ(x) involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of f_λ(x) is dense in [0,1/2], pose a few open questions, and formulate a conjecture relating the integral to the length of the partition. We also provide specific examples throughout to support our speculation that an in-depth analysis of partition polynomials could further strengthen our understanding of partitions.