Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois. However, certain classes of quintic equations can be solved in this manner. Irreducible quintic equations can be associated with a Galois group, which may be a symmetric group S_n, metacyclic group M_n, dihedral group D_n, alternating group A_n, or cyclic group C_n, as illustrated above. Solvability of a quintic is then predicated by its corresponding group being a solvable group.

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