## Quick Reference

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

Theorem

Every polynomial equation

*a*_{n}*z*^{n}+*a*_{n−1}*z*^{n−1}+⋯+*a*_{1}*z*+*a*_{0}=0,

where the *a*_{i} are real or complex numbers and *a*_{n} ≠ 0, has a root in the set of complex numbers.

It follows that, if *f*(*z*)=*a*_{n}*z*^{n}+*a*_{n−1}*z*^{n−1}+…+*a*_{1}*z*+*a*_{0}, there exist complex numbers α_{1}, α_{2},⋯,α_{n} (not necessarily distinct) such that

*f*(*z*)=*a*_{n}(*z*−α_{1})(*z*−α_{2})⋯(*z*−α_{n}).

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

*Subjects:*
Mathematics.

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