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2.5 complex 0’s & the
fundamental theorem of algebra
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).
• Example: find the complex conjugate
1. 4 – 3i
2. 8 + 2i
3. 3 + √3 i
Factoring a complex polynomial
function
• Use the Rational Root theorem
Examples: find all zeros
1. f(x) = 3x4 + 8x3 + 6x2 + 3x – 2
2. f(x) = x3 – 10x2 + 44x – 69
Fundamental theorem of algebra
A polynomial of degree n has n complex roots
(real and non-real). Some of these may be
repeated.
In other words: if you can solve the polynomial
the polynomial has 0’s
For example: if k is a zero of a polynomial if and
only if x – k is a factor of the polynomial
How to use the fundamental theorem
of algebra
• 2 parts: 1) find all roots, how you do this depends on if you are
given a real root or a complex root
a) given a real root: do synthetic division, rewrite your
answer, find the remaining zeros by doing quadratic formula
(these will be complex numbers),write your answer in set
notation.)
b) given a complex root: find the conjugate, write both as
(x – (a + bi))(x – (a – bi)) and foil to get rid of the i’s, take that
quadratic and the original problem and find the remaining roots
by doing long division. If need be factor or use quadratic
formula to find the remaining roots or set equal
to 0 and solve, write your answer in set notation.
Cont’d
• 2) given the roots find the polynomial function
write your roots as (x - #)(x - #),etc. If given a
complex root or a radical, DO NOT forget about
it’s conjugate. Multiply the parenthesis together
either by FOIL or distributing. Write your final
answer as f(x) = …
Use the fundamental theorem of
algebra
• Use the fundamental theorem of algebra to find
all zeros
• Examples:
1. f(x) = x3 – 3x2 + 9x + 13; 2 – 3i
2. f(x) = x3 + 6x2 + 21x + 26; -2
Finding a polynomial given zeros
• **if there is an i or radical in 1 parenthesis you
must have the conjugate in another
• Examples: write a polynomial with minimal
degree in standard form with real coefficients
whose zeros include those listed. Use the
fundamental theorem of algebra
1. -4, 1 – i, and 1 + i
2. -2, 1- i
3. -2- i, 1 + 3i
Examples cont’d
4. -1 (multiplicity 3), 3 (multiplicity 1)
(x – 1)3(x – 3) = (x – 1)(x – 1)(x – 1)(x – 3)
5. -1 (multiplicity 2), -2 – i (multiplicity 1)
Fundamental Polynomial connections
in the complex case
• The following statements about a polynomial
function f are equivalent if k is a complex
number:
1. x = k is a solution (or root) of the equation
f(x) = 0
2. k is a zero of the function f
3. x – k is a factor of f(x)
Write the function in std. form &
identify the 0’s and x-intercepts of the
graph
Examples:
1) f(x) = (x – 2i)(x + 2i)
2) f(x) = (x – 5)(x - √2 i)(x + √2 i)
3) f(x) = (x – 1)(x – 1)(x + 2i)(x – 2i)