A Newman polynomial has all the coefficients in {0,1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {-1,1}. We call P(X) in Z[X] a Borwein polynomial if all its coefficients belong to {-1,0,1} and P(0) not equal to 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z|=1 has a non-zero multiple in Z[X] with coefficients in a finite set D subset Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.