3.7 Complex Zeros 2011
October 17, 2011
3.7 ­ Complex Zeros
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Objectives: Use the conjugate pairs theorem.
Find a polynomial when the zeros are given.
Find zeros when some are already given.
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Warm-up:
1. Find k such that f(x) = x3 ­ kx2 + kx + 2 has the factor x ­ 2.
2. What is the remainder when f(x) = 2x20 ­ 8x10 + x ­ 2 is divided by x ­ 1? 2
3.7 Complex Zeros 2011
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Review:
Find the real zeros of the polynomial function and write the function in factored form.
f(x) = 2x3 + 3x2 + 2x + 3
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Polynomial Function
Degree Zeros
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*Note: Multiple zeros are counted more than once
AND complex zeros are counted.
Complex Zero: A number consisting of a real part and an imaginary part, written: a + bi .
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0 Conjugate Pairs Theorem
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Complex zeros of polynomial
0 functions occur in conjugate pairs.
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If a + bi is a zero of f, 0then the complex conjugate a ­ bi is also a real zero of f.
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Example: If 3 ­ i is a zero, then 3 + i must also be a zero.
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Example #1: 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
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Example #2: 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 ­ 215x + 204
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Example #3: 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
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Example #4: 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
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Example #5
Use the given zero to find the remaining zeros.
f(x) = x4 ­ 7x3 + 13x2 ­ 7x + 12
zero: ­i
Step 1: Determine the number of zeros.
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Step 2: Find the complex conjugate of the given zero.
­i, i
Step 3: Write the zeros as factors and expand.
(x ­ i)(x + i)
x2 + ix ­ ix ­ i2
x2 ­ (­1)
x2 + 1
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Example #5
Use the given zero to find the remaining zeros.
f(x) = x4 ­ 7x3 + 13x2 ­ 7x + 12
zero: ­i
Step 4: Use long division to find the remaining zeros.
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x ­ 7x + 13x ­ 7x + 12
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Example #5
Use the given zero to find the remaining zeros.
f(x) = x4 ­ 7x3 + 13x2 ­ 7x + 12
zero: ­i
Step 5: Factor if necessary to find remaining zeros.
zeros: ­ i, i, 3, 4
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Example #6
Use the given zero to find the remaining zeros.
f(x) = x4 ­ 7x3 + 19x2 ­ 23x + 10
zero: 2 ­ i
zeros: 2 ­ i, 2 + i, 1, 2
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Homework
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page 237 0
(3 ­ 28) skip #20
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