Download Lesson 4.5

The Fundamental
Theorem of Algebra
Lesson 4.5
Example

2
x
 3x  5  0
Consider the solution to
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Note the graph
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No intersections
with x-axis
Using the
solve and
csolve
functions
Fundamental Theorem of
Algebra
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A polynomial f(x) of degree n ≥ 1 has at least
one complex zero
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Number of Zeros theorem
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Remember that complex includes reals
A polynomial of degree n has at most n distinct zeros
Explain how theorems apply to these graphs
Constructing a Polynomial
with Prescribed Zeros
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Given polynomial f(x)
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Degree = 4
Leading coefficient 2
Zeros -3, 5, i, -i
Determine factored form
f ( x)  2   x  3 x  5 x  i  x  i 
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Determine expanded form
Conjugate Zeroes Theorem
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Given a polynomial with real coefficients
P( x)  an x  an 1 x
n
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n 1
 ...  a1 x  a0
If a + bi is a zero, then a – bi will also be a
zero
Finding Imaginary Zeros
f ( x)  x  x  2 x  x  1
4
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Given
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Determine all zeros
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Use calculator to factor
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Try cFactor command
Use calculator to graph
Use cSolve or cZeros
3
2
Application
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Complex numbers show up in study of electrical
circuits
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Impedance, Z
Voltage, V
Current, I
Can be represented by
complex numbers
Find missing value
V
Z
I
V  30  60i I  8  6i
Z  10  5i V  10  8i
Assignment
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Lesson 4.5
Page 307
Exercises 1 – 41 EOO
43 – 47 odd