3.5 Complex Zeros and the Fundamental Theorem of Algebra

Fundamental theorem of algebra

Every polynomial with complex coefficient has at least one complex zero

Complete factorization theorem

If P(x) is a polynomial of degree n≥1, then there exist complex numbers a, c1, c2,...,cn (a≠0)such that P(x)=a(x-c1)(x-c2)...(x-cn)

Multiplicity

If the factor x-c appears k times in the complex factorization of P(x), then we say that c is a zero of _________ k.

Conjugate zero theorem

If the polynomial P has real coefficients and if the complex number z is a zero of P, the then the complex conjugate is also a zero of P.

Irreducible

A quadratic polynomial with no real zeros is called _________ over the real numbers