COMPLEX ZEROS

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SECTION 3.6

• COMPLEX ZEROS
• FUNDAMENTAL THEOREM OF ALGEBRA

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COMPLEX POLYNOMIAL FUNCTION
A complex polynomial function f of degree n is
a complex function of the form f(x) a n x n a
n-1 x n-1 . . . a1x a0 where an, a n-1, . .
., a1, a0 are complex numbers, an ? 0, n is a
nonnegative integer, and x is a complex variable.
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COMPLEX ZERO
A complex number r is called a complex zero of a
complex function f if f(r) 0.
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COMPLEX ZEROS
We have learned that some quadratic equations
have no real solutions but that in the complex
number system every quadratic equation has a
solution, either real or complex.
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FUNDAMENTAL THEOREM OF ALGEBRA
Every complex polynomial function f(x) of degree
n ? 1 has at least one complex zero.
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THEOREM
Every complex polynomial function f(x) of degree
n ? 1 can be factored into n linear factors (not
necessarily distinct) of the form f(x) an(x -
r1)(x - r2)? ? ? (x - rn) where an, r1, r2, . .
., rn are complex numbers.
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CONJUGATE PAIRS THEOREM
Let f(x) be a complex 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 zero of f.
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CONJUGATE PAIRS THEOREM
In other words, for complex polynomials whose
coefficients are real numbers, the zeros occur in
conjugate pairs.
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CORORLLARY
A complex polynomial f of odd degree with real
coefficients has at least one real zero.
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EXAMPLE
A polynomial f of degree 5 whose coefficients are
real numbers has the zeros 1, 5i, and 1 i.
Find the remaining two zeros. - 5i 1 - i
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EXAMPLE
Find a polynomial f of degree 4 whose
coefficients are real numbers and has the zeros
1, 1, and - 4 i. f(x) a(x - 1)(x - 1)x - (-
4 i)x - (- 4 - i) First, let a 1 Graph
the resulting polynomial. Then look at other as.
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EXAMPLE
It is known that 2 i is a zero of f(x) x4 -
8x3 64x - 105 Find the remaining zeros. - 3, 7,
2 i and 2 - i
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EXAMPLE
Find the complex zeros of the polynomial
function f(x) 3x4 5x3 25x2 45x - 18
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• CONCLUSION OF SECTION 3.6