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October 08, 2014
Section 2.5
Complex Zeros and the
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
A polynomial function of degree n has n
complex zeros. Some of these zeros may
be repeated.
Linear Factorization Theorem
These statements are equivalent.
1. x = k is a solution (or root) to the equation
2. k is a zero of the function f.
3. k is an x-intercept of the graph of
4. x − k is a factor of
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October 08, 2014
Complex Conjugate Zeros
Suppose that f (x) is a polynomial function
with real coefficients. If a and b are real
numbers with b ≠ 0 and a + bi is a zero of
f (x), then its complex conjugate a − bi is also
a zero of f (x).
Suppose 3 − 7i is a zero to f (x) = 0. Name
another zero. Write a quadratic function with
these two zeros.
Consider
[x - (3 + 7i)] [x - (3 - 7i)] =
[x - 3 - 7i] [x - 3 + 7i] =
[(x - 3) - 7i] [(x - 3) + 7i] =
Remember (a - b) (a + b) = a2 - b2
(x - 3)2 - (7i)2 =
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October 08, 2014
Find all the zeros of the function
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A quadratic expression with real coefficients
but no real zeros is called an irreducible factor.
Write
as a product
of linear and irreducible quadratic factors
all with real coefficients.
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Page 215 #35
Suppose 3 - 2i is one of the root
of f(x) = x4 - 6x3 + 11x2 + 12x - 26.
Find the linear factorization of f.
f(x)=[x-(3-2i)][x-(3+2i)](x-√2)(x+√2)
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