We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach one. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a non-linear, low-degree factor. We also study characteristic polynomials of random matrices with +1 and −1 entries.We study many polynomial models that appear to satisfy Heuristic 1.1 (primarily with independent coefficients, though some with dependence, see Section 4), and the "well-behaved" condition is meant to exclude models with specific features that cause high-degree factors, for example, a random polynomial of degree d formed as the product of two random polynomials with degree around d/2. Throughout, "reducible" will be used as shorthand for "reducible over the rationals." Note that for monic polynomials, reducibility over the rationals is equivalent to reducibility over the integers by Gauss's Lemma. For non-monic polynomials, we are interested in cases where all Key words and phrases. random integer polynomials, irreducible, low-degree factors.

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