9. Fundamental Theorem
of Algebra
Every polynomial equation P(x) = 0 of any degree
with complex coefficients has at least one complex
number solution.
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11. Double Root
In a quadratic, when the discriminant equals 0,
there will be two roots that have the same value.
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12. Double Root
In a quadratic, when the discriminant equals 0,
there will be two roots that have the same value.
When any root appears twice.
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14. Multiplicity of a Root
The highest power of (x - r) of a polynomial when r is
a root.
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15. The Number of Roots
of a Polynomial
Equation Theorem
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16. The Number of Roots
of a Polynomial
Equation Theorem
Every polynomial of degree n has exactly n roots
(including multiplicity).
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17. Example 1: How many
roots does each
equation have?
4 2
a. x15 +1=0 b. 2x − 3x + π = 0
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18. Example 1: How many
roots does each
equation have?
4 2
a. x15 +1=0 b. 2x − 3x + π = 0
15
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19. Example 1: How many
roots does each
equation have?
4 2
a. x15 +1=0 b. 2x − 3x + π = 0
4
15
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20. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
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21. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2:
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22. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2
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23. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2
q = factors of 1:
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24. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2
q = factors of 1: ±1
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25. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
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26. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
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27. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
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28. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
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29. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
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30. Example 2:
Consider the equation x4 - 6x3 + 2x2 - 3x + 2 = 0.
a. How many real solutions are there?
p = factors of 2: ±1,±2 p
= ±1,±2
q
q = factors of 1: ±1
Wait a minute! Why isn’t the zero
of this function either 1 or 2?
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31. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
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32. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
Wednesday, March 18, 2009
33. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
Wednesday, March 18, 2009
34. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
Wednesday, March 18, 2009
35. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
Wednesday, March 18, 2009
36. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
With this being a quartic (4th degree), we also know that
there should be 3 changes in the curvature. Are there?
Wednesday, March 18, 2009
37. The rational-root theorem only tells us where to look for
rational roots. What does this tell us about the roots of
our function?
So let’s graph on the following window:
-5 ≤ x ≤ 10 and -110 ≤ y ≤ 40
With this being a quartic (4th degree), we also know that
there should be 3 changes in the curvature. Are there?
There are 2 real solutions.
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38. b. How many roots (real or complex) are there?
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39. b. How many roots (real or complex) are there?
x4 - 6x3 + 2x2 - 3x + 2 = 0
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40. b. How many roots (real or complex) are there?
x4 - 6x3 + 2x2 - 3x + 2 = 0
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41. b. How many roots (real or complex) are there?
x4 - 6x3 + 2x2 - 3x + 2 = 0
4
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42. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
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43. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2
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44. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2
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45. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x
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46. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x + 25) = 0
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47. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x + 25) = 0
x2(x + 5)2 = 0
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48. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x + 25) = 0
x2(x + 5)2 = 0
x = 0, -5
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49. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x + 25) = 0
x2(x + 5)2 = 0
x = 0, -5
What can we say about these roots?
Wednesday, March 18, 2009
50. Example 3:
Find all roots of 4 + 10x3 + 25x2 =0
x
x2(x2 + 10x + 25) = 0
x2(x + 5)2 = 0
x = 0, -5
What can we say about these roots?
They are each double roots.
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