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2023-02-20

How to determine the number of complex roots of a polynomial of degree n?

Bobby Espinoza

Beginner2023-02-21Added 7 answers

For example, Fundamental Theorem of Algebra

Any non-constant polynomial in one variable with Complex (potentially Real) coefficients has a zero in C, according to the Fundamental Theorem of Algebra (FTOA) (the set of Complex numbers). This has the simple implication that a polynomial of degree n with Complex (possibly Real) coefficients has precisely n Complex (possibly Real) zeros counting multiplicity, which is frequently expressed as part of the FTOA. A polynomial of degree n thus has precisely n Complex zeros counting multiplicity, which is a straightforward answer to your query. Which n zeros - if any - are Real and which aren't? Any Complex zeros that exist will be found in pairs of Complex conjugates if the polynomial contains Real coefficients. Thus, there will be an even number of non-real zeros.

Any non-constant polynomial in one variable with Complex (potentially Real) coefficients has a zero in C, according to the Fundamental Theorem of Algebra (FTOA) (the set of Complex numbers). This has the simple implication that a polynomial of degree n with Complex (possibly Real) coefficients has precisely n Complex (possibly Real) zeros counting multiplicity, which is frequently expressed as part of the FTOA. A polynomial of degree n thus has precisely n Complex zeros counting multiplicity, which is a straightforward answer to your query. Which n zeros - if any - are Real and which aren't? Any Complex zeros that exist will be found in pairs of Complex conjugates if the polynomial contains Real coefficients. Thus, there will be an even number of non-real zeros.