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Good, okay, so now with this being said we're 
going to talk about another theorem in a minute,

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that's basically this proof, almost identically,

3
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but it's a different setting. So it's not 
just over any field in the next setting,

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00:00:12,900 --> 00:00:14,966
but it'll be over the complex numbers.

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But to get there, we need a hammer 
and the hammer that we're going to

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talk about is the Fundamental 
Theorem of Algebra.

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The Fundamental Theorem 
of Algebra, what does it say?

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It says that every non-constant 
polynomial has a complex root.

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That's a very, very 
simple-to-write statement, 

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but it's not so easy to prove.

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Every non-constant complex 
polynomial has a complex root.

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So if you remember where we've been 
going with this polynomial section,

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we argued that okay well we have,

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you know, all these polynomials 
in different rings over

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integral polynomials, rational 
polynomials, real polynomials,

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but still there are polynomials

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that don't have roots. For 
example x squared plus 1. 

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And we said, “Oh well let's go 
down this complex number route

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and talk about complex numbers and 
say okay, well complex numbers...

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help us to solve polynomials 
like x squared plus 1,”

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and you could probably try to reason to 
yourself that okay well maybe all these

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real polynomials they must have complex roots.

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That's true, but

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maybe one thing that's not entirely 
clear, and really this isn't clear at all,

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but why do all complex polynomials 
also have to have complex roots?

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Neither of those two statements are very obvious 
actually, but it turns out that they're true.

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Every non-constant polynomial has a 
complex root of some sort, okay?

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So I can always factor 
complex polynomials.

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That's weird, right, we added this 
element i to the real numbers 

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and linear combinations 
of it, and we somehow

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argue that hey well by 
doing this, we actually get…

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00:01:41,000 --> 00:01:43,700
we can actually factor every 
polynomial we encounter

36
00:01:43,700 --> 00:01:46,166
over this new field C.

37
00:01:46,166 --> 00:01:47,000


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00:01:47,000 --> 00:01:50,600
We call C algebraically closed sometimes.

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00:01:50,600 --> 00:01:53,300
Not important, but it's good 
to know a little bit of words.

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Knowledge is power, right, somebody
told me that once upon a time.

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00:01:56,333 --> 00:01:58,433
Okay, great.

42
00:01:58,433 --> 00:02:00,866
So here we go. We're not going 
to prove this, the proof is hard.

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00:02:00,866 --> 00:02:04,366
I find that the easiest proof for me to 
understand is in complex analysis,

44
00:02:04,366 --> 00:02:06,700
but that takes a little bit to get to. 
So once you have the right tools

45
00:02:06,700 --> 00:02:09,333
t's not too hard to prove this, but 
we don't have anywhere near

46
00:02:09,333 --> 00:02:11,633
the correct amount of tools to prove this.

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00:02:11,633 --> 00:02:14,366
I have seen a proof of 
this that's elementary, but

48
00:02:14,366 --> 00:02:18,100
it's tough to understand, it really 
is, without the correct tools.

49
00:02:18,100 --> 00:02:22,800
There's also a proof in Galois Theory which, 
as a course, I recommend completely. It's a...

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00:02:22,800 --> 00:02:26,133
it's a wonderful course. Galois Theory is 
one of my favorite undergraduate courses

51
00:02:26,133 --> 00:02:28,766
and if you get a chance to 
take it, you absolutely should.

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00:02:28,766 --> 00:02:32,733
You talk about things like this, the Fundamental 
Theorem of Algebra, you discuss things

53
00:02:32,733 --> 00:02:36,666
like why degree 2, 3, 
and 4 polynomials

54
00:02:36,666 --> 00:02:38,466
they all have

55
00:02:38,466 --> 00:02:41,433
nice formulas that can be 
used - or that can be…like

56
00:02:41,433 --> 00:02:43,800
the quadratic formula 
and Cardano’s formula,

57
00:02:43,800 --> 00:02:46,400
those all exists for degree 2, 
3, and 4 polynomials, but

58
00:02:46,400 --> 00:02:49,466
degree 5 and higher, you can't 
find some nice polynomial using

59
00:02:49,466 --> 00:02:52,233
addition, subtraction, 
square roots, etc.

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00:02:52,233 --> 00:02:53,733


61
00:02:53,733 --> 00:02:57,666
It's great, it's mind-blowing, it's a beautiful 
subject you should take it if you get a chance.

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00:02:57,666 --> 00:02:59,466
If not it's a shame, but

63
00:02:59,466 --> 00:03:02,933
let's plug away. I've already talked 
about this theorem more than I want to.

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00:03:02,933 --> 00:03:05,433
I love this theorem, I think it's pretty cool, but

65
00:03:05,433 --> 00:03:08,733
that’s okay let's actually try to use 
it to prove something, okay?

66
00:03:08,733 --> 00:03:09,666


67
00:03:09,666 --> 00:03:12,666
I guess before I do that, the polynomial x 
squared plus 1 over the real numbers

68
00:03:12,666 --> 00:03:16,166
shows that this doesn't 
happen over all fields, okay?

69
00:03:16,166 --> 00:03:20,033
So x squared plus 1 factors 
over C, it has complex roots, but

70
00:03:20,033 --> 00:03:24,800
x squared plus 1 doesn't have real roots, 
okay, that's what I want to try to say here.

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00:03:24,800 --> 00:03:27,033


72
00:03:27,033 --> 00:03:30,666
So something very special 
with the complex numbers.

73
00:03:30,666 --> 00:03:35,733
It's quite magical. The more you think about it, 
the more you think, “Wow why is this true?”

74
00:03:35,733 --> 00:03:37,033
Anyways...

75
00:03:37,033 --> 00:03:38,333


76
00:03:38,333 --> 00:03:41,966
let's use the Fundamental 
Theorem of Algebra to prove this

77
00:03:41,966 --> 00:03:45,700
statement: Complex Polynomials 
of Degree n Have n Roots.

78
00:03:45,700 --> 00:03:47,966
What is this saying? Well if I 
have a complex polynomial,

79
00:03:47,966 --> 00:03:52,000
then I can factor it completely 
as follows: I can write this as 

80
00:03:52,000 --> 00:03:56,633
some constant c, which is going to be 
the leading coefficient of z to the n,

81
00:03:56,633 --> 00:04:01,633
and I can split it up as z minus c1, z minus 
c2, all the way down to z minus c n.

82
00:04:01,633 --> 00:04:04,000
So every complex polynomial 
can be factored like this.

83
00:04:04,000 --> 00:04:04,833


84
00:04:04,833 --> 00:04:06,766
These roots might not 
necessarily be distinct,

85
00:04:06,766 --> 00:04:09,766
but they are roots of the polynomial f at z.

86
00:04:09,766 --> 00:04:12,533
So for example, I mean 
just kind of visualize it.

87
00:04:12,533 --> 00:04:15,066
I didn't actually write down the 
roots here, but the polynomial

88
00:04:15,066 --> 00:04:18,000
2z to the 7, plus z to 
the 5, plus i z, plus 7

89
00:04:18,000 --> 00:04:21,000
can be written in this form, that's 
what this theorem is saying.

90
00:04:21,000 --> 00:04:25,466
What this theorem doesn't tell you, and what’s 
actually quite difficult, is what are these roots?

91
00:04:25,466 --> 00:04:26,233


92
00:04:26,233 --> 00:04:28,466
Factoring is a hard, hard problem.

93
00:04:28,466 --> 00:04:30,333
We've already seen this with integers,

94
00:04:30,333 --> 00:04:34,766
it's not much different with polynomials, 
it's still pretty challenging.

95
00:04:34,766 --> 00:04:36,166


96
00:04:36,166 --> 00:04:41,800
Sorry about that, I got distracted. There was 
something in the background, I closed it down.

97
00:04:41,800 --> 00:04:43,000


98
00:04:43,000 --> 00:04:45,033
So let's talk about this theorem,

99
00:04:45,033 --> 00:04:47,000
let's talk about actually 
trying to prove this, okay?

100
00:04:47,000 --> 00:04:50,900
Complex Polynomials of 
Degree n Have n Roots.

101
00:04:50,900 --> 00:04:53,233
How do we prove this? Well it turns 
out that the proof is going to be

102
00:04:53,233 --> 00:04:55,566
very similar to the proof of

103
00:04:55,566 --> 00:04:56,500


104
00:04:56,500 --> 00:05:01,833
there being only n roots of a polynomial 
of degree n over a field F, okay?

105
00:05:01,833 --> 00:05:05,000
The difference here is that…well you'll 
see where the difference is, okay?

106
00:05:05,000 --> 00:05:06,766


107
00:05:06,766 --> 00:05:11,266
So how do we prove this statement? We're going 
to prove this by mathematical induction again.

108
00:05:11,266 --> 00:05:15,233
When n equals 1, it's similar to before. We 
have some a z plus b, we're going to write it - 

109
00:05:15,233 --> 00:05:17,933
so we're going to factor out the a, 
we're going to use a little bit of 

110
00:05:17,933 --> 00:05:21,100
negation magic, and we're 
going to write this as a times

111
00:05:21,100 --> 00:05:23,066
z minus negative b over a. 

112
00:05:23,066 --> 00:05:26,366
This is valid because a is not 
0, hence is invertible in C.

113
00:05:26,366 --> 00:05:28,233


114
00:05:28,233 --> 00:05:31,366
And that's good, so this satisfies 
the form that we have.

115
00:05:31,366 --> 00:05:33,833
The induction hypothesis, 
we’re going to assume

116
00:05:33,833 --> 00:05:37,700
that all polynomials of degree k over the 
complex numbers can be written in this form,

117
00:05:37,700 --> 00:05:40,533
and the inductive conclusion 
well how is this going to work?

118
00:05:40,533 --> 00:05:41,833


119
00:05:41,833 --> 00:05:44,600
And the idea is that we do need to use 
the Fundamental Theorem of Algebra

120
00:05:44,600 --> 00:05:48,266
because what does it tell us? Well Fundamental 
Theorem of Algebra gives us a factor,

121
00:05:48,266 --> 00:05:51,533
right? We know that we must have a root 
by the Fundamental Theorem of Algebra.

122
00:05:51,533 --> 00:05:54,266
The Factor Theorem tells us 
that we must have a factor.

123
00:05:54,266 --> 00:05:57,133
So we know that z minus c1, let's say,

124
00:05:57,133 --> 00:06:00,133
is a factor of our polynomial. Okay? 

125
00:06:00,133 --> 00:06:03,133
Oh I guess I've called it c k plus 1. 

126
00:06:03,133 --> 00:06:04,100


127
00:06:04,100 --> 00:06:06,800
So z minus c k plus 1 
is a factor of f at z,

128
00:06:06,800 --> 00:06:10,300
so I can write f at z as z minus 
c k plus 1, times g of z,

129
00:06:10,300 --> 00:06:12,533
where g of z has degree k,

130
00:06:12,533 --> 00:06:15,033
and just like before now I 
have a degree k polynomial,

131
00:06:15,033 --> 00:06:18,900
so by the inductive hypothesis, 
I can write it out as

132
00:06:18,900 --> 00:06:22,733
c times z minus c1 all 
the way up to z minus c k.

133
00:06:22,733 --> 00:06:26,500
I combine it together, I get f at z is 
equal to this product of polynomials

134
00:06:26,500 --> 00:06:29,600
from i equals 1 to k plus 1 of z minus c i,

135
00:06:29,600 --> 00:06:32,233
and that's exactly what I wanted to prove.

136
00:06:32,233 --> 00:06:36,600
This proof is almost identical to that 
proof before. The thing that’s changed

137
00:06:36,600 --> 00:06:41,100
is that I don’t have a 0 root case. I always have 
a root by the Fundamental Theorem of Algebra.

138
00:06:41,100 --> 00:06:44,000
That's the difference between this proof 
and the proof from three slides ago.

139
00:06:44,000 --> 00:06:44,833


140
00:06:44,833 --> 00:06:47,566
So something to think about there. 

141
00:06:47,566 --> 00:06:49,166


142
00:06:49,166 --> 00:06:53,166
Okay, really cool, really neat. I like that 
you can reuse that theorem from before,

143
00:06:53,166 --> 00:06:56,833
that proof from before, I think it's pretty 
cool. But it does help us out here. 

144
00:06:56,833 --> 00:06:58,433


145
00:06:58,433 --> 00:07:02,533
So great, so we know how to factor - well we know 
that there is a factorization of complex polynomials.

146
00:07:02,533 --> 00:07:05,666
How to find that factorization, 
you need to use things like

147
00:07:05,666 --> 00:07:08,000
the Rational Roots Theorem,

148
00:07:08,000 --> 00:07:10,833
and the Factor Theorem, and

149
00:07:10,833 --> 00:07:14,033
and the Remainder Theorem, all these sorts of 
theorems, and the Conjugate Roots Theorem.

150
00:07:14,033 --> 00:07:16,666
But in particular, we're going 
to head to those last two:

151
00:07:16,666 --> 00:07:19,333
the Rational Root Theorem and 
the Conjugate Roots Theorem.

152
00:07:19,333 --> 00:07:21,299
They'll help us to factor polynomials.