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Fundamental theorem of algebra

The fundamental theorem of algebra reads:

The properties of complex numbers mean that a polynomial of degree n has exactly n roots in the complex numbers (instead of at most n roots as for real numbers).

Also, the equation xn = a has a solution for all n unequal 0, for negative numbers a, instead of just for odd values of n.

 


Explanation

Any complex polynomial in the single variable z factorizes into linear factors

a0 + a1z + a2z2 + ··· + anzn = an(z − b1)(z − b2) ··· (z − bn)

 


Example 1

The equation x2 + i = 0 has two solutions

because

 


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