copyright © 2012 pearson education, inc. publishing as prentice hall. section 5.6 complex zeros;...

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

Section 5.6

Complex Zeros;

Fundamental Theorem of Algebra


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.


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: 1 1 4 4f x a x x x i x i

2 22 1 4 4 4 4a x x x i x i x i i

2 2 22 1 4 4 16 4 4a x x x x ix x ix i i i

2 22 1 8 17a x x x x

4 3 210 34 42 17a x x x x


4 3 2Find the complex zeros of the polynomial function and write in factored form.

2 8 20f

f x x x x x

Step 1: The degree of f is 4 so there will be 4 complex zeros.

The potential rational zeros are : 1, 2, 4, 5, 10, 20pq

Step 2:

3 2 2 4 9 10f x x x x x

2 1 4 9 10 2 4 10

1 2 5 0

2 1 2 1 8 20 2 8 18 20

1 4 9 10 0

2 2 2 2 5f x x x x x

22 0 or 2 0 or 2 5 0x x x x

22 2 2 5 0x x x x

22 2 4 1 52 or 2 or

2 1x x x


4 3 2Find the complex zeros of the polynomial function and write in factored form.

2 8 20f

f x x x x x 2 2 2 2 5f x x x x x

22 2 4 1 52 or 2 or

2 1x x x

2 16 1 22

i

The four complex zeros of are 2,2, 1 2 , 1 2 .f i i

2 2 1 2 1 2f x x x x i x i

copyright © 2012 pearson education, inc. publishing as prentice hall. section 5.6 complex zeros;...

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