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We are asked to show that the ideal $(x, 5)$ is not a principal ideal in $\mathbb{Z}[x]$, i.e., the ring of polynomials with integer coefficients. --- ### Step 1: Understanding the Ideal $(x, 5)$ The ideal $(x, 5)$ in $\mathbb{Z}[x]$ consists of all elements of the form: $$ f(x) = a(x) \cdot x + b(x) \cdot 5 = a(x)x + 5b(x) $$ for $a(x), b(x) \in \mathbb{Z}[x]$. So any _______ ___ _________ _____ _______ __________ __________ ___ ___ ______ ________ ____.
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