# 3_6 - zeros occur as conjugate pairs • Fact 2 A...

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Section 3.6: Complex zeros of Polynomials Instructor: Ms. Hoa Nguyen 1 Recall Section 3.4 and Section 3.5 Section 3.4 : Every polynomial of real coeﬃcients a n , a n - 1 , ··· , a 1 , a 0 can be fac- tored into either LINEAR factors or IRREDUCIBLE QUADRATIC factors. An irreducible quadratic factor is a quadratic function whose discriminant ( b 2 - 4 ac ) is NEGATIVE . Section 3.5 : If a quadratic function of real coeﬃcients has a NEGATIVE discriminant, then the zeros are a complex (NOT REAL) number z and its complex conjugate ¯ z . In general : Fact 1 : If a polynomial of REAL coeﬃcients has a complex zero, then the complex conjugate of this zero is also a zero of this polynomial. In other words, the complex

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Unformatted text preview: zeros occur as conjugate pairs. • Fact 2 : A polynomial of degree n ≥ 1 can always be factored into n linear factors, if complex numbers are allowed in the linear factors. Section 3.4: The corollary of the Intermediate Value Theorem If f is a polyno-mial of ODD degree, then f has at least ONE real zero. Question : How can you prove the corollary, using the above two facts of the polynomial of REAL coeﬃcients? 2 The Fundamental Theorem of Algebra A polynomial of degree n ≥ 1 has at least one COMPLEX zero. Remember that a real number is a complex number with the imaginary part equal to 0. Example 1 1 Example 2 2...
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3_6 - zeros occur as conjugate pairs • Fact 2 A...

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