Complex Zeros Fundamental Theorem of Algebra

1
Complex Zeros Fundamental Theorem of Algebra
2
A complex polynomial function f degree n is a
complex function of the form
Where are complex numbers.
A complex number r is called a (complex) zero of
a complex function f if f (r) 0.
3
Fundamental Theorem of Algebra
Every complex polynomial function f (x) of degree
n gt 1 has at least one complex zero.
4
Fundamental Theorem of Algebra
Every complex polynomial function f (x) of degree
n gt 1 can be factored into n linear factors (not
necessarily distinct) of the form
5
Find the zeros ofUse the zeros to factor f

• According to the quadratic formula

6
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.
7
Find a polynomial f of degree 4 whose
coefficients are real numbers and that has zeros
1, 2, and 2i.
f(x)
8
Given where all the coefficients are real.

• What is the maximum number of real zeros that f
  can have?
• What is the minimum number of zeros that f can
  have?
• What is the maximum number of complex (but not
  real) zeros of f?

9
Find the complex zeroes of the polynomial function
There are 4 complex zeros.
From Rational Zero Theorem find potential
rational zeros
10
(No Transcript)
11
Zeros are -2, -1/2, 2 5i, 2 -5i.