Download Complex Zeroes Intro

Complex Zeros; Fundamental
Theorem of Algebra
A complex polynomial function f degree n is a
complex function of the form
Where a0 , a1 ,...an are complex
numbers.
A complex number r is called a (complex) zero
of a complex function f if f (r) = 0.
Fundamental Theorem of Algebra
Every complex polynomial function f (x) of
degree n > 1 has at least one complex zero.
Fundamental Theorem of Algebra
Every complex polynomial function f (x) of
degree n > 1 can be factored into n linear
factors (not necessarily distinct) of the form
Find the zeros of f ( x)  x 2  4 x  5
Use the zeros to factor f
According to the quadratic formula
4  (4)  4(1)( 5)
x
2(1)
2
4   4 4  2  1 4  2i



 2i
2
2
2
f ( x )  ( x  ( 2  i ))( x  ( 2  i ))
 ( x  2  i )( x  2  i )
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 is also a zero of f.
Corollary
A complex polynomial f of odd degree with real
coefficients has at least one real zero.
Find a polynomial f of degree 4 whose
coefficients are real numbers and that has zeros
1, 2, and 2+i.
f(x)
Given f ( x )  a5 x  a4 x   a0
where all the coefficients are real.
5
4
a) What is the maximum number of real
zeros that f can have?
b) What is the minimum number of zeros
that f can have?
c) What is the maximum number of complex
(but not real) zeros of f?
Find the complex zeroes of the polynomial function
There are 4 complex zeros.
From Rational Zero Theorem find potential
rational zeros
Zeros are -2, -1/2, 2 +5i, 2 -5i.