In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a computation of $B_{
2n}$ as a function of B$_{2n-2}$ only. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k $in mathbb Z$ is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of $zeta$(3), $zeta$(5) and $zeta$(7).
