Suppose given a smooth morphism $f:X\to Y$ between varieties over $\mathbb{C}$ whose fibres are $\mathbb{P}^n$. Then I have an equality of Hodge polynomials $H(X) = H(Y)H(\mathbb{P}^n)$, say because the hyperplane class generates the cohomology of $\mathbb{P}^n$ and hence $f_* \mathbb{Z}_X$ cannot have monodromy.

Is any such fibration in fact Zariski locally trivial? Even if not, do I have the equality in the Grothendieck group of varieties $[X] = [Y][\mathbb{P}^n]$?