3.7 ­ Complex Zeros
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Objective: To use the conjugate pairs theorem
To find a polynomial when the zeros are given
Title: Dec 2 ­ 11:15 AM (1 of 8)
A polynomial function of degree n will always have exactly n complex zeros. These zeros may be comprised of real, imaginary or complex numbers Title: Dec 2 ­ 11:44 AM (2 of 8)
Conjugate pairs theorem
Let f(x) be a polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate r = a ­ bi is also a real zero of f.
Example: If 3 ­ i is a zero, then 3 + i must also be a zero
Title: Dec 2 ­ 11:18 AM (3 of 8)
Using the given information, find the remaining zeros of f.
Degree 4; zeros: 3, 4, 4 ­ i
4 + i
Degree 4; zeros: i, 1 + i
­i, 1 ­ i
Degree 5; zeros: 1, ­i, 2 + i
i, 2 ­ i
Title: Dec 2 ­ 11:16 AM (4 of 8)
Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 4; zeros: 3, 4, 4 ­ i
f(x) = (x ­ 3)(x ­ 4)(x ­(4 ­ i))(x ­ (4 + i))
f(x) = x4 ­ 15x3 + 85x2 ­ 215 x + 204
Title: Dec 2 ­ 11:16 AM (5 of 8)
Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 4; zeros: i, 1 + i
f(x) = (x ­ i)(x + i)(x ­ (1 + i))(x ­ (1 ­ i))
f(x) = x4 ­ 2x3 + 3x2 ­ 2x + 2
Title: Dec 2 ­ 11:16 AM (6 of 8)
Form a polynomial f(x) with real coefficients having the given degree and zeros.
Degree 5; zeros: 1, ­i, 2 + i
f(x) = x5 ­ 5x4 + 10x3 ­ 10x2 + 9x ­ 5
Title: Dec 2 ­ 11:16 AM (7 of 8)
Homework: pg: 237
#'s: 3­6, 8­16 evens, 17­19
Title: Dec 2 ­ 11:33 AM (8 of 8)