Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to $\mu$ while ensuring that $f(0)\neq0$, then there is a positive constant $\theta=\theta(\mu)$ such that $f(x)$ has no divisors of degree $\le \theta n$ with probability that tends to 1 as $n\to\infty$. Furthermore, in certain cases, we show that a random polynomial $f(x)$ with $f(0)\neq0$ is irreducible with probability tending to 1 as $n\to\infty$. In particular, this is the case if $\mu$ is the uniform measure on a set of at least 35 consecutive integers, or on a subset of $[-H,H]\cap\mathbb{Z}$ of cardinality $\ge H^{4/5}(\log H)^2$ with $H$ sufficiently large. In addition, in all of these settings, we show that the Galois group of $f(x)$ is either $\mathcal{A}_n$ or $\mathcal{S}_n$ with high probability. Finally, when $\mu$ is the uniform measure on a finite arithmetic progression of at least two elements, we prove a random polynomial $f(x)$ as above is irreducible with probability $\ge\delta$ for some constant $\delta=\delta(\mu)>0$. In fact, if the arithmetic progression has step 1, we prove the stronger result that the Galois group of $f(x)$ is $\mathcal{A}_n$ or $\mathcal{S}_n$ with probability $\ge\delta$.