Download Radical, Irrational, Too Complex to be Real!

The Fundamental
Theorem of Algebra
Intro - Chapter
4.6
The Fundamental Theorem of Algebra
Every non-constant polynomial has a ______
ZERO in the
complex number system.
MOST
• Every polynomial of degree n > 0 has AT
__________
ZEROS in the complex number system.
n different _______
• If every zero is counted as many times as its
MULTIPLICITY then, a polynomial of degree n has
______________
__________
EXACTLY n complex zeros.
Example 6: Find a polynomial f  x of degree 3 such
that 1, and – 2 are zeros, 1 has a multiplicity of 2 and
f  2  32
f  x   a  x  1  x  2 
2
 f  x   8  x  1  x  2 
2
32  a  2  1  2  2 
32  4a
a 8
2
Conjugate Zero Theorem
REAL coefficients.
Let f  x  be a polynomial with ______
If the complex number z is a zero of f  x  ,then its
CONJUGATE
_____________, ____, is also a zero of f  x  .
z
Example 7: Find a polynomial with real coefficients
whose zeros include the numbers 2 and 3+ i
a  x  2  x  3  i   x  3  i 
let a  1
 x  2   x   3  i  x   3  i  x   9  1 
2
 x  2   x  3x  ix  3x  ix  10 
2
3
2
x

2
x

6
x

10


  x  8 x  22 x  20
2
Example 8: Factor f x   x 4  5x 3  4 x 2  2 x  8
All possible zeros:  1, 2, 4, 8
4
1 5

4 2 8
4 4 0 8
1 1
 x  4  x
0 2 0
3
x 2
2
2  2  4  2
2
x

2
Complex plane:
1 1 1 0
2
  1 2 2
1 2 2
Real plane:
0

f  x    x  4  x  1 x 2  2 x  2
2  4 2  2i


 1 i
2
2
f  x    x  4  x  1  x  1  i    x  1  i  
