  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).
