3.5 Complex Zeros & the Fundamental Theorem of Algebra.

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Presentation transcript:

3.5 Complex Zeros & the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra An nth degree polynomial has exactly n zeros in the complex number system.

Find all the zeros and factor completely. P(x) = x 3 + x x + 81 Final Possibilities Total Zeros

Find all the zeros and factor completely. P(x) = x 4 - 3x 3 + 7x x - 26 Final Possibilities Total Zeros

Find all the zeros and factor completely. P(x) = 3x x x Total Zeros

Find the polynomial with zeros at i, -i, 2, -2 and P(5)=273

Complex Conjugate Theorem If a + bi is a root then a – bi is also a root.

Find the polynomial with zeros at:

Use Descartes’ Rule to count the number of real and imaginary zeros. P(x) = x 3 – 100x x + 50 Two changes in sign: 2 or 0 positive real zeros One change in sign: 1 negative real zeros Positive Real ZerosNegative Real ZerosImaginary Zeros

Every polynomial with real coefficients can be factored into the product of linear and irreducible quadratic factors with real coefficients. Factor f(x) = x 4 + 9x 2 – 112 into: Linear and irreducible quadratic factors Linear factors with complex coefficients Linear and Quadratic Factors Theorem

pg 298 #9, 11, odd