1
00:00:05,000 --> 00:00:12,020
Let us factor this expression. To factor means to write it as a product.

2
00:00:12,520 --> 00:00:16,640
What do we call this polynomial?

3
00:00:18,920 --> 00:00:20,120
It is a quadratic binomial.

4
00:00:25,280 --> 00:00:28,960
Is it missing the quadratic, linear, or constant term?

5
00:00:33,080 --> 00:00:35,600
It is missing the linear term.

6
00:00:36,820 --> 00:00:38,940
We could write it like this.

7
00:00:44,420 --> 00:00:53,340
We need a factor pair whose product is negative 49 and whose sum is zero.

8
00:00:55,420 --> 00:00:56,500
The pair is negative seven and positive seven.

9
00:00:56,500 --> 00:00:57,400
plus seven.

10
00:01:02,460 --> 00:01:04,060
That's plus seven.

11
00:01:04,740 --> 00:01:07,720
That gives x minus seven and x plus seven.

12
00:01:09,560 --> 00:01:11,860
Yes. Easy peasy. Wow.

13
00:01:13,800 --> 00:01:14,700
Okay.

14
00:01:16,000 --> 00:01:19,700
Do we need to write this step? Definitely not.

15
00:01:19,700 --> 00:01:22,160
We need two numbers whose product is negative five and whose sum is zero.

16
00:01:22,160 --> 00:01:29,480
to be negative 5 and add to be 0.

17
00:01:34,460 --> 00:01:36,480
Maybe it's not.

18
00:01:37,380 --> 00:01:40,600
Maybe it's not.

19
00:01:41,280 --> 00:01:43,840
Definitely not.

20
00:01:44,400 --> 00:01:45,840
What did we...

21
00:01:56,720 --> 00:01:58,360
Somebody said... What did somebody say?

22
00:01:59,500 --> 00:02:00,400
What did you say?

23
00:02:02,380 --> 00:02:02,900
Uh...

24
00:02:02,900 --> 00:02:03,140
Rad...

25
00:02:03,140 --> 00:02:04,460
Use square root of five and negative square root of five.

26
00:02:04,680 --> 00:02:05,660
I'm gonna be a bad man.

27
00:02:06,760 --> 00:02:07,280
Uh...

28
00:02:07,280 --> 00:02:08,620
Rad pod and negative red pod.

29
00:02:13,840 --> 00:02:19,380
Those numbers multiply to negative five and add to zero.

30
00:02:21,440 --> 00:02:24,080
These factors are not integers, but they are valid real numbers.

31
00:02:25,300 --> 00:02:30,288
Integers, but they are two numbers that multiplied to be negative five and add to be

32
00:02:30,288 --> 00:02:31,038
zero

33
00:02:32,560 --> 00:02:37,640
The factorization is x minus square root of five times x plus square root of five.

34
00:02:39,820 --> 00:02:42,840
Should we check and see if it works?

35
00:02:44,140 --> 00:03:00,320
Multiply to check: the middle terms cancel, and the product is x squared minus five.

36
00:03:01,100 --> 00:03:02,980
Oops, I normally do it here.

37
00:03:02,980 --> 00:03:06,980
here. And we get minus 5

38
00:03:10,000 --> 00:03:11,060
at 0

39
00:03:11,060 --> 00:03:12,180
and that's it.

40
00:03:16,080 --> 00:03:16,600
Okay.

41
00:03:16,600 --> 00:03:24,840
Can two real numbers multiply to 36 and add to zero?

42
00:03:31,700 --> 00:03:33,120
It's not 6 and negative 6.

43
00:03:33,260 --> 00:03:34,160
Why not?

44
00:03:36,000 --> 00:03:38,820
Yeah, that does not multiply to be 36.

45
00:03:42,780 --> 00:03:43,680
Are there any?

46
00:03:46,120 --> 00:03:47,020
There are none.

47
00:03:48,660 --> 00:03:53,479
They would need the same sign to have a positive product, so their sum cannot be

48
00:03:53,479 --> 00:03:54,229
zero.

49
00:03:55,700 --> 00:03:59,356
Two positives add together, never will be zero. Two negatives add together will

50
00:03:59,356 --> 00:04:00,200
never be zero.

51
00:04:01,040 --> 00:04:02,740
Therefore this expression does not factor over the real numbers.

52
00:04:08,900 --> 00:04:12,140
This is called a sum of squares.

53
00:04:15,860 --> 00:04:18,260
A sum of squares does not factor over the real numbers.

54
00:04:18,260 --> 00:04:18,460
Okay.

55
00:04:18,460 --> 00:04:21,260
The earlier examples are differences of squares.

56
00:04:21,440 --> 00:04:22,340
What do we call this?

57
00:04:22,820 --> 00:04:23,720
The difference of squares.

58
00:04:23,960 --> 00:04:24,860
Difference.

59
00:04:28,160 --> 00:04:30,600
Oh, squares.

60
00:04:34,100 --> 00:04:34,680
Okay.

61
00:04:34,680 --> 00:04:39,600
A difference of two squares always factors.

62
00:04:44,020 --> 00:04:47,620
A squared minus b squared factors as a minus b times a plus b.

63
00:04:47,620 --> 00:04:51,120
always factors to what?

64
00:04:54,240 --> 00:04:56,600
A plus B times

65
00:04:56,600 --> 00:04:59,700
A minus B.

66
00:04:59,700 --> 00:05:04,820
and that order doesn't matter.

67
00:05:06,900 --> 00:05:08,580
Let's call the

68
00:05:08,580 --> 00:05:10,040
difference of squares.

69
00:05:33,900 --> 00:05:43,400
Write x squared minus one hundred as x squared minus ten squared.

70
00:05:43,400 --> 00:05:48,180
It factors as x minus ten times x plus ten.

71
00:05:48,420 --> 00:05:48,440
OK.

72
00:05:48,440 --> 00:05:52,840
Rather than multiplying everything out, recognize the square terms.

73
00:05:52,840 --> 00:05:54,840
and then look for two factors and...

74
00:05:56,920 --> 00:05:57,820
Too complicated.

75
00:05:58,680 --> 00:05:59,580
It's very easy.

76
00:06:00,700 --> 00:06:01,800
Four x squared is the square of two x.

77
00:06:03,520 --> 00:06:04,420
2x.

78
00:06:05,120 --> 00:06:06,580
Thirty-six is the square of six.

79
00:06:07,240 --> 00:06:08,140
Six.

80
00:06:09,840 --> 00:06:11,360
This is a difference of squares.

81
00:06:11,760 --> 00:06:12,680
It factors as two x minus six times two x plus six.

82
00:06:13,080 --> 00:06:14,720
2x minus six.

83
00:06:15,500 --> 00:06:17,080
That's 2x plus six.

84
00:06:17,720 --> 00:06:17,880
Done.

85
00:06:17,880 --> 00:06:23,580
Let's try another one.

86
00:06:37,660 --> 00:06:40,120
This expression is a difference of squares.

87
00:06:45,960 --> 00:06:50,540
One hundred x squared is the square of ten x, and eighty-one is the square of nine.

88
00:06:50,540 --> 00:06:54,000
81 is 9.

89
00:06:56,080 --> 00:07:00,340
It factors as ten x minus nine times ten x plus nine.

90
00:07:03,860 --> 00:07:06,660
plus 9

91
00:07:13,300 --> 00:07:14,700
Uh...

92
00:07:25,980 --> 00:07:29,440
The square roots are four x and five y.

93
00:07:30,100 --> 00:07:36,600
So the factors are four x minus five y and four x plus five y.

94
00:07:37,080 --> 00:07:37,200
Done.

95
00:07:37,200 --> 00:07:44,300
Okay, let's try another one.

96
00:08:00,060 --> 00:08:01,200
This is also a difference of squares.

97
00:08:01,440 --> 00:08:02,720
The square root of this?

98
00:08:08,920 --> 00:08:11,780
The square root of twenty-five x squared is five x.

99
00:08:12,000 --> 00:08:14,280
What's the square root of this?

100
00:08:14,280 --> 00:08:25,140
The square root of seven stays square root of seven, giving conjugate factors.

101
00:08:25,140 --> 00:08:27,620
This is a fourth-degree binomial rather than a quadratic binomial.

102
00:08:27,740 --> 00:08:29,760
What would this be?

103
00:08:29,760 --> 00:08:31,900
4th degree bound.

104
00:08:38,260 --> 00:08:41,780
However, it is still a difference of squares.

105
00:08:48,440 --> 00:08:50,392
The square root of x to the fourth is x squared, and the square root of sixteen is

106
00:08:50,392 --> 00:08:50,960
four.

107
00:08:50,960 --> 00:08:52,740
x squared.

108
00:08:53,440 --> 00:08:54,440
What squared is 16?

109
00:08:55,440 --> 00:08:55,820
4.

110
00:08:55,820 --> 00:08:59,860
4. This is going to be

111
00:08:59,860 --> 00:09:03,760
First factor as x squared minus four times x squared plus four.

112
00:09:03,760 --> 00:09:05,580
squared plus 4.

113
00:09:08,640 --> 00:09:10,760
We are not finished because x squared minus four is another difference of squares.

114
00:09:13,340 --> 00:09:14,360
Factor x squared minus four again.

115
00:09:14,360 --> 00:09:16,780
difference of squares.

116
00:09:19,720 --> 00:09:29,376
The complete factorization is x minus two times x plus two times x squared plus

117
00:09:29,376 --> 00:09:30,126
four.

118
00:09:32,340 --> 00:09:39,280
The sum of squares x squared plus four is prime over the real numbers.

119
00:09:41,380 --> 00:09:48,400
Difference of squares lets us factor higher-degree polynomials.

120
00:09:54,160 --> 00:09:57,740
And it's going to be very useful when we learn something else later.

121
00:09:57,740 --> 00:09:59,400
But let's try one more.

122
00:10:06,840 --> 00:10:08,700
This is another difference of squares.

123
00:10:15,000 --> 00:10:17,420
What square is x need?

124
00:10:19,240 --> 00:10:22,440
The square root of x to the eighth is x to the fourth, and the square root of one is

125
00:10:22,440 --> 00:10:23,190
one.

126
00:10:32,220 --> 00:10:38,440
And we have two fourth degree polynomials. Is one of the two difference the squares?

127
00:10:38,440 --> 00:11:04,563
Factor x to the fourth minus one again as x squared minus one times x squared plus

128
00:11:04,563 --> 00:11:06,100
one.

129
00:11:07,820 --> 00:11:09,620
We are not done because x squared minus one is another difference of squares.

130
00:11:10,800 --> 00:11:12,380
No. We got another

131
00:11:14,100 --> 00:11:15,040
difference of squares.

132
00:11:29,140 --> 00:11:31,740
Now the expression is completely factored.

133
00:11:35,140 --> 00:11:36,040
What's that?

134
00:11:37,680 --> 00:11:39,460
The result is the product of four binomials, and their order does not matter.

135
00:11:39,460 --> 00:11:42,340
4 binomials. The order does not matter.

136
00:11:43,580 --> 00:11:47,100
Because the multiplication is commuted. Sorry, let's do one more.

137
00:11:47,100 --> 00:11:49,040
First factor out the greatest common factor.

138
00:11:51,660 --> 00:11:52,560
Factor out what?

139
00:11:53,660 --> 00:11:57,600
Always remove the greatest common factor before applying another factoring method.

140
00:11:58,800 --> 00:12:00,180
What is the greatest common factor?

141
00:12:05,700 --> 00:12:07,820
The greatest common factor is ten x cubed.

142
00:12:15,980 --> 00:12:21,240
The remaining quadratic binomial is a difference of squares.

143
00:12:42,660 --> 00:12:44,460
Okay, we good?