Summary: We consider random polynomials whose coefficients are independent and uniform on \(\{-1,1\}\). We prove that the probability that such a polynomial of degree \(n\) has a double root is \(o(n^{-2})\) when \(n+1\) is not divisible by 4 and asymptotic to \(\frac{8\sqrt 3}{\pi n^2}\) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on \(\{-1,0,1\}\) and whose largest atom is strictly less than \(1/\sqrt 3\). In this general case, we prove that the probability of having a double root equals the probability that either \(-1\), \(0\) or \(1\) are double roots up to an \(o(n^{-2})\) factor and we find the asymptotics of the latter probability.