Fundamental Theorem of Algebra

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Notes 6.6 Fundamental Theorem of Algebra
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Presentation transcript:

Fundamental Theorem of Algebra Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

A polynomial of degree 5 whose coefficients are real numbers has the zeros 2, 3i, and 2 + 4i. Find the remaining two zeros. Since f has coefficients that are real numbers, complex zeros appear as conjugate pairs. It follows that 3i, the conjugate of 3i, and 2  4i, the conjugate of 2 + 4i, are the two remaining zeros. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 1, 1, 4+i. Since 4 + i is a zero, by the Conjugate Pairs Theorem, 4  i is also a zero. By the factor theorem: Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Step 1: The degree of f is 4 so there will be 4 complex zeros. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

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