The Fundamental Theorem of Algebra

Lesson 4.5

Example

 Consider the solution to

x

2  3

x

 Note the graph  No intersections with x-axis  Using the

solve

and

csolve

functions 0

Fundamental Theorem of Algebra

 A polynomial f(x) of degree n ≥ 1 has at least one complex zero  Remember that complex includes reals 

Number of Zeros theorem

 A polynomial of degree

n

has at most

n

distinct zeros  Explain how theorems apply to these graphs

Constructing a Polynomial with Prescribed Zeros

 Given polynomial f(x)    Degree = 4 Leading coefficient 2 Zeros -3, 5,

i, -i

 Determine factored form



x

3



x

 5

  

 Determine expanded form

Conjugate Zeroes Theorem

 Given a polynomial with real coefficients 

a x n n



a n

 1

x n

 1

a x

1 

a

0  If

a + bi

zero is a zero, then

a – bi

will also be a

Finding Imaginary Zeros

 Given 

x

4 

x

3  2

x

2  Determine all zeros  Use calculator to factor  Try

cFactor

command   Use calculator to graph Use

cSolve

or

cZeros

1

Application

 Complex numbers show up in study of electrical circuits    Impedance,

Z

Voltage,

V

Can be represented by Current,

I

complex numbers

Z



V I

 Find missing value

Z



V

  

i I

 

i i V

 

i

Assignment

 Lesson 4.5

  Page 307 Exercises 1 – 41 EOO 43 – 47 odd