Polynomial optimization is a special case of dc (Difference of Convex functions) programming, however representing a multivariate polynomial into a dc function is a difficult task. We propose some new results on dc programming formulations for polynomial optimization. We are interested in polynomial decomposition techniques for representing any multivariate polynomial into difference-of-sums-of-squares (DSOS) and difference-of-convex-sums-of-squares (DCSOS) polynomials. We first prove that the sets of DSOS and DCSOS polynomials are vector spaces equivalent to the set of real valued polynomials. We also show that the problem of finding DSOS and DCSOS decompositions is equivalent to semidefinite programs (SDPs). Then, we focus on establishing several practical algorithms for DSOS and DCSOS decompositions without solving SDPs. Some examples illustrate how to use our methods.