3.8 Complex Zeros; Fundamental Theorem of Algebra

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Presentation transcript:

3.8 Complex Zeros; Fundamental Theorem of Algebra

A variable in the complex number system is referred to as a complex variable. A complex polynomial function f degree n is a complex function of the form A complex number r is called a (complex) zero of a complex function f if f (r) = 0.

Fundamental Theorem of Algebra Every complex polynomial function f (x) of degree n > 1 has at least one complex zero.

Theorem Every complex polynomial function f (x) of degree n > 1 can be factored into n linear factors (not necessarily distinct) of the form

Conjugate Pairs Theorem Let f (x) be a complex polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate is also a zero of f. Corollary A complex polynomial f of odd degree with real coefficients has at least one real zero.

Find a polynomial f of degree 4 whose coefficients are real numbers and that has zeros 1, 2, and 2+i. f(x)

Find the complex zeroes of the polynomial function There are 4 complex zeros. From Rational Zero Theorem find potential rational zeros

Zeros are -2, -1/2, 2 +5i, 2 -5i.