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**dori1123** Let $\displaystyle f(x)$ be a polynomial in $\displaystyle \mathbb{Z}[x]$. Prove that if $\displaystyle f(x)$ has a root $\displaystyle \alpha \in \mathbb{Z}$, then $\displaystyle f(x)$ has a linear factor in $\displaystyle \mathbb{Z}[x]$.

Suppose $\displaystyle f(x)$ has a root $\displaystyle \alpha \in \mathbb{Z}$, then $\displaystyle f(x)$ is reducible, and then $\displaystyle f(x) = a(x)(x - \alpha)$ for some nonconstant polynomial $\displaystyle a(x) \in \mathbb{Z}[x]$. So $\displaystyle x - \alpha$ is a linear factor in $\displaystyle \mathbb{Z}[x]$.

Is this correct? Did I miss anything important in this proof?