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Could you provide me an example of a polynomial that is not uniquely factorized in $\mathbb{Z}+x \mathbb{Q}[x]$? Thanks!

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We have $x^2=(x/5)\cdot 5x$ and $x^2=x\cdot x$. The key point being $5x$ and $x$ aren't associates because $5$ is not a unit in $\mathbb Z +x\mathbb Q[x]$.