The fundamental theorem of algebra was first stated by d’Alembert in 1746 but was only partially proved. Gauss, at the age of twenty-one, was the first to give a rigorous proof of the theorem. In fact he gave three alternate proofs to the theorem, and all three proofs can be found in the third volume of his Works.

The fundamental theorem of algebra states that every equation of the n th degree has n roots. We can alternatively express this theorem as: the polynomial

can always be factoreded into n linear factors of the form z – a _i .

The proof of this theorem is done in two steps, first showing that an equation of the n th degree has at least one roots and then showing that the equation has n roots and no more.

Note that it is possible for several of the n roots a₁, a_2,, . . . ,a_n to be the same. e.g. a₁ = a₂ = a₃. In this case a₁ is a multiple root and for the example given is a root of multiplicity three.

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