= Solution
The <Hilbert basis theorem> makes $k[x_1,\ldots,x_n]$ a <Noetherian ring>. Reversing the inclusion of ideals shows that its <Zariski-closed set>[Zariski-closed subsets] satisfy the descending-chain condition, so every affine variety is a <Noetherian Zariski topology>[Noetherian topological space].
Suppose some nonempty closed subset had no finite decomposition into <irreducible topological space>[irreducible] closed subsets. It would be reducible, so it could be written
$$
V=V_1\cup V_2
$$
with $V_1,V_2$ proper closed subsets. At least one of these would again admit no finite irreducible decomposition. Repeating that choice would produce an infinite strictly descending chain of closed subsets, contradicting Noetherianity. Therefore
$$
\boxed{V\text{ is a finite union of irreducible subvarieties}.}
$$
Removing any member contained in another leaves exactly the <irreducible component>[irreducible components].
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