Obj: To use the Fundamental Theorem of
Algebra to solve polynomial equations with
complex roots
We have solved polynomial equations and found their roots
are included in the set of complex numbers.
Recall: Complex Numbers =
Real #s + Imaginary #s
Therefore, our roots have been:
-integers
-rational numbers
-irrational numbers
-imaginary numbers
But, can all polynomial equations be solved using
complex numbers?
Carl Friedrich Gauss (1777-1855)
says roots of every
polynomial equation, even
those with imaginary
coefficients, are complex
numbers.
--
--developed the
Fundamental Theorem of
Algebra
Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n ≥ 1 with complex
coefficients, then P(x) = 0 has at least one complex
root.
Corollary
Including imaginary roots and multiple roots, as nth
degree polynomial equation has exactly n roots
Another way of saying this
is: you can factor a
polynomial of degree n into
n linear factors.
Imaginary Root Theorem
If the imaginary number a +bi is a root of a polynomial
equation with real coefficients, then the conjugate
a- bi is also a root.
Therefore, imaginary roots always
_________________________.
Using the Fundamental Theorem of Algebra:
Example 1: For the equation x3 + 2x2 – 4x – 6 = 0, find:
a. Number of complex roots:
b. Number of real roots:
c. Possible rational roots:
Example 2: For the equation x4 – 3x3 + x2 – x + 3 = 0,
find:
a. Number of complex roots:
b. Number of real roots:
c. Possible rational roots:
Finding all zeros of a Polynomial Function:
Example 3: Find the number of complex zeros of
f(x) = x5 + 3x4 – x – 3. FIND ALL ZEROS.
Example 4: Find the number of complex zeros of
f(x) =x3 – 2x2 +4x – 8. FIND ALL ZEROS.
CLOSURE:
How can you determine how many roots there
are for a polynomial equation?
Find the
degree
 Pg
337 # 1-8, 9-15 odd, 21,23