1
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2
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Multiply and express as a simplified rational.

3
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State the domain.

4
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Let's multiply it, and then before we simplify it, let's

5
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look at the domain.

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This is equal to, if we just multiplied the numerators, a

7
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squared minus 4 times a plus 1, all of that over-- multiply

8
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the denominators-- a squared minus 1 times a plus 2.

9
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10
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Now, the a squared minus 4 and the a squared minus 1 might

11
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look familiar to us.

12
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These are the difference in squares, a special type of

13
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binomial that you could immediately, or hopefully

14
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maybe immediately recognize.

15
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It takes the form a squared minus b squared, difference of

16
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squares, and it's always going to be equal to a plus b

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times a minus b.

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We can factor this a squared minus 4 and we can also factor

19
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this a squared minus 1, and that'll help us actually

20
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simplify the expression or simplify the rational.

21
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This top part, we can factor the a squared minus 4 as a

22
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plus 2-- 2 squared is 4-- times a minus 2, and then that

23
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times a plus 1.

24
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25
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Then in the denominator, we can factor a squared minus 1--

26
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let me do that in a different color.

27
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a squared minus 1 we can factor as a plus 1

28
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times a minus 1.

29
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If you ever want to say, hey, why does this work?

30
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Just multiply it out and you'll see that when you

31
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multiply these two things, you get that thing right there.

32
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Then in the denominator, we also have an a plus 2.

33
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34
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We've multiplied it, we've factored out the numerator, we

35
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factored out the denominator.

36
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Let's rearrange it a little bit.

37
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So this numerator, let's put the a plus 2's first in both

38
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the numerator and the denominator.

39
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So this, we could get a plus 2 in the numerator, and then in

40
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the denominator, we also have an a plus 2.

41
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In the numerator, we took care of our a plus 2's.

42
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That's the only one that's common, so in the numerator,

43
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we also have an a minus 2.

44
00:02:15,199 --> 00:02:16,689
Actually, we have an a plus 1-- let's

45
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write that there, too.

46
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We have an a plus 1 in the numerator.

47
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We also have an a plus 1 in denominator.

48
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49
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In the numerator, we have an a minus 2, and in the

50
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denominator, we have an a minus 1.

51
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So all I did is I rearranged the numerator and the

52
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denominator, so if there was something that was of a

53
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similar-- if the same expression was in both, I just

54
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wrote them on top of each other, essentially.

55
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Now, before we simplify, this is a good time to think about

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the domain or think about the a values that aren't in the

57
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domain, the a values that would invalidate or make this

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expression undefined.

59
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Like we've seen before, the a values that would do that are

60
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the ones that would make the denominator equal 0.

61
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So the a values that would make that equal to 0 is a is

62
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equal to negative 2.

63
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You could solve for i.

64
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You could say a plus 2 is equal to 0, or a is equal to

65
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negative 2.

66
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a plus 1 is equal to 0.

67
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Subtract 1 from both sides. a is equal to negative 1.

68
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Or a minus 1 is equal to 0.

69
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Add one to both sides, and you get a is equal to 1.

70
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For this expression right here, you have to add the

71
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constraint that a cannot equal negative 2, negative 1, or 1,

72
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that a can be any real number except for these.

73
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We're essentially stating our domain.

74
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We're stating the domain is all possible a's except for

75
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these things right here, so we'd have to add that little

76
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caveat right there.

77
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Now that we've done that, we can factor it.

78
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We have an a plus 2 over an a plus 2.

79
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We know that a is not going to be equal to negative 2, so

80
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that's always going to be defined.

81
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When you divide something by itself, that is

82
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going to just be 1.

83
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The same thing with the a plus 1 over the a plus 1.

84
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That's going to be 1.

85
00:04:01,330 --> 00:04:03,790
All you're going to be left with is an a minus 2

86
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over a minus 1.

87
00:04:05,590 --> 00:04:11,200
So the simplified rational is a minus 2 over a minus 1 with

88
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the constraint that a cannot equal negative 2,

89
00:04:15,789 --> 00:04:18,079
negative 1, or 1.

90
00:04:18,079 --> 00:04:20,610
You're probably saying, Sal, what's wrong with it equaling,

91
00:04:20,610 --> 00:04:21,970
for example, negative 1 here?

92
00:04:21,970 --> 00:04:25,145
Negative 1 minus 1, it's only going to be a negative 2 here.

93
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It's going to be defined.

94
00:04:26,459 --> 00:04:30,689
But in order for this expression to really be the

95
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same as this expression up here, it has to have the same

96
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constraints.

97
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It has to have the same domain.

98
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It cannot be defined at negative 1 if this guy also is

99
00:04:42,449 --> 00:04:43,930
not defined at negative 1.

100
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And so these constraints essentially ensure that we're

101
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dealing with the same expression, not one that's

102
00:04:49,069 --> 00:04:50,740
just close.

103
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