Let \(k\) be a field of characteristic \(p > 0\), and let \(f\in k[x]\) be irreducible. Then \(f\) is inseparable if and only if \(f(x) = g(x^p)\) for some \(g\in k[x]\). Moreover, there are a unique integer \(n \geq 0\) and an irreducible separable polynomial \(g\in k[x]\) with \(f(x) = g(x^{p^n})\), and \(f\) is inseparable if and only if \(n \geq 1\).
A polynomial has a repeated root in a splitting field if and only if it shares a root with its derivative. Since \(f\) is irreducible, \(f\) shares a root with \(f'\) if and only if \(f \divides f'\), and since \(\deg f' < \deg f\) this holds if and only if \(f' = 0\). Writing \(f = \sum_i a_i x^i\), we have \(f' = \sum_i i a_i x^{i-1}\), which vanishes if and only if \(a_i = 0\) whenever \(p \nmid i\), that is, if and only if \(f(x) = g(x^p)\) for some \(g\in k[x]\).
For the second statement, let \(n\) be the largest integer with \(f\in k[x^{p^n}]\); it exists because \(\deg f \geq 1\). Write \(f(x) = g(x^{p^n})\). A factorization of \(g\) into nonconstant polynomials would give one of \(f\), so \(g\) is irreducible, and \(g\notin k[x^p]\) by maximality of \(n\), so \(g\) is separable by the first statement. Conversely, if \(f(x) = g(x^{p^m})\) with \(g\) separable and irreducible, then \(g\notin k[x^p]\) by the first statement, so \(f\notin k[x^{p^{m+1}}]\) and \(m = n\). Finally, \(n\geq 1\) if and only if \(f\in k[x^p]\), which by the first statement holds if and only if \(f\) is inseparable.