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How many complex roots does a polynomial equation of degree n have?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

According to the fundamental theorem of algebra, a polynomial equation of degree n with complex coefficients has exactly n roots in the complex number system.12

It is important to note that these roots are not necessarily distinct. When counting the total number of roots, one must account for their multiplicity. For example, a quadratic equation (degree 2) such as $x^{2} - 2 x + 1 = 0$ can be factored as $(x - 1) (x - 1) = 0$ , meaning the root $x = 1$ has a multiplicity of 2.1

While the theorem guarantees that an equation of degree n has exactly n complex roots, it does not imply that these roots will always be real numbers, even if the coefficients of the polynomial are real. If the coefficients are real and the degree n is odd, the polynomial is guaranteed to have at least one real root.3