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Fundamental Theorem of Algebra


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The following important theorem in mathematics, concerned with the roots of polynomial equations:

Theorem

Every polynomial equation

anzn+an−1zn−1+⋯+a1z+a0=0,

where the ai are real or complex numbers and an ≠ 0, has a root in the set of complex numbers.

It follows that, if f(z)=anzn+an−1zn−1+…+a1z+a0, there exist complex numbers α1, α2,⋯,αn (not necessarily distinct) such that

f(z)=an(z−α1)(z−α2)⋯(z−αn).

Hence the equation f(z)=0 cannot have more than n distinct roots.

Subjects: Mathematics.


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