A Newman polynomial is an element of \(\mathbb{Z}[x]\) with coefficients in \(\{0,1\}\) and constant term \(1.\) A Littlewood polynomial is one with coefficients in \(\{-1,1\}.\) Let \(V_{\mathcal{N}}\) be the set of roots of Newman polynomials, \(V_{\mathcal{L}}\) the roots of Littlewood polynomials, and \(V\) the roots of polynomials with coefficients in \(\{-1,0,1\}\) with nonzero constant term. The authors study the relationships amongst \(V_{\mathcal{N}},\) \(V_{\mathcal{L}}\), and \(V.\) In particular, they show that \(V_{\mathcal{N}}\) is not a subset of \(V_{\mathcal{L}}\) (and hence \(V_{\mathcal{N}},\) \(V_{\mathcal{L}},\) and \(V\) are distinct sets) by giving explicit examples of irreducible Newman polynomials that do not divide any Littlewood polynomial; e.g., \(x^9+x^6+x^2+x+1\). They also show that there are no such Newman polynomials of degree less than or equal to \(8\). On the other hand, the authors prove that every Newman trinomial \(1+x^a+x^b\) divides some Littlewood polynomial. A similar result is obtained for certain quadrinomials as well. The proofs are mostly computational. Two numerical algorithms are given: one for checking whether a polynomial in \(\mathbb{Z}[x]\) can divide a Littlewood polynomial, and one that determines whether a given Newman polynomial \(P\) divides a Littlewood polynomial \(L\) such that the height of \(L/P\) is less than or equal to a given integer.