Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 5.6 Complex Zeros;...
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Transcript of 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
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: 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
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.
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
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
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