Download 3.6 - San Juan College

SECTION 3.6
COMPLEX ZEROS;
FUNDAMENTAL THEOREM OF
ALGEBRA
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.
COMPLEX ZERO
A complex number r is called a complex
zero of a complex function f if f(r) = 0.
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.
FUNDAMENTAL THEOREM
OF ALGEBRA
Every complex polynomial function f(x)
of degree n  1 has at least one complex
zero.
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.
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.
CONJUGATE PAIRS
THEOREM
In other words, for complex
polynomials whose coefficients are
real numbers, the zeros occur in
conjugate pairs.
CORORLLARY
A complex polynomial f of odd
degree with real coefficients has at
least one real zero.
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
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 a’s.
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
EXAMPLE
Find the complex zeros of the
polynomial function
f(x) = 3x4 + 5x3 + 25x2 + 45x - 18
CONCLUSION OF SECTION 3.6