1
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2
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All right, we're on problem 44.

3
00:00:02,589 --> 00:00:07,740
And they said, which is the factored form of 3a squared

4
00:00:07,740 --> 00:00:15,129
minus 24ab plus 48b squared?

5
00:00:15,130 --> 00:00:18,085
And if this confuses you with a's and b's instead of an x

6
00:00:18,085 --> 00:00:21,750
there, I just like to think of in this case, this looks like

7
00:00:21,750 --> 00:00:22,559
an x squared.

8
00:00:22,559 --> 00:00:24,129
So I like to think of it the same way I would think of it

9
00:00:24,129 --> 00:00:27,730
in terms of if this was an x squared minus some number

10
00:00:27,730 --> 00:00:29,740
times x plus some other number.

11
00:00:29,739 --> 00:00:31,279
So let's put that in the back of your head.

12
00:00:31,280 --> 00:00:33,250
If that's confuses you, don't worry about it.

13
00:00:33,250 --> 00:00:35,229
But the first thing I like to do is try to get rid of the

14
00:00:35,229 --> 00:00:38,809
coefficient on the, in this case, the a squared term.

15
00:00:38,810 --> 00:00:40,730
Let's see if we can factor out a number.

16
00:00:40,729 --> 00:00:41,250
Well, sure.

17
00:00:41,250 --> 00:00:42,729
All of these are divisible by 3.

18
00:00:42,729 --> 00:00:45,589
3 is divisible by 3, 24 is divisible by 3 and 48 is

19
00:00:45,590 --> 00:00:46,230
divisible by 3.

20
00:00:46,229 --> 00:00:48,809
So we should be able to factor out a 3, so let's do that.

21
00:00:48,810 --> 00:00:55,484
So that is equal to 3 times a squared minus 8ab.

22
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23
00:00:58,130 --> 00:00:59,900
24 divided by 3 is 8 and there's a

24
00:00:59,899 --> 00:01:01,170
minus in front of it.

25
00:01:01,170 --> 00:01:04,799
Plus 48 divided by-- 3.

26
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My brain was getting ahead of me.

27
00:01:06,109 --> 00:01:11,590
48 divided by 3 is 16b squared.

28
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Now to factor this we have to think about it as, are there

29
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two numbers-- well, think of it.

30
00:01:16,849 --> 00:01:18,209
Let me rewrite this a little bit.

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00:01:18,209 --> 00:01:20,179
If we view a as kind of the independent variable-- you

32
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wouldn't have to do it, but I just wanted you to visualize

33
00:01:22,349 --> 00:01:23,369
it properly.

34
00:01:23,370 --> 00:01:32,960
3a squared minus 8b times a plus 16b squared.

35
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So if we viewed a squared as kind of the independent

36
00:01:35,239 --> 00:01:38,559
variable or the x term, so now this kind of has the shape of

37
00:01:38,560 --> 00:01:40,900
polynomials that hopefully you're used to factoring a

38
00:01:40,900 --> 00:01:41,490
little bit.

39
00:01:41,489 --> 00:01:46,169
We just have to think, OK, are there two numbers that add up

40
00:01:46,170 --> 00:01:51,590
to minus 8b, and that when I multiply the two numbers, I

41
00:01:51,590 --> 00:01:55,150
get 16b squared?

42
00:01:55,150 --> 00:01:56,330
So first all, the number is going to be

43
00:01:56,329 --> 00:01:57,469
in terms of b, obviously.

44
00:01:57,469 --> 00:02:00,590
Because if I'm adding them and I get a b and-- I get b's.

45
00:02:00,590 --> 00:02:02,299
And if I square them I get a b squared.

46
00:02:02,299 --> 00:02:04,599
And it also makes me lead to it's the same number, just

47
00:02:04,599 --> 00:02:05,299
used twice.

48
00:02:05,299 --> 00:02:09,728
Otherwise, it would be weird to get a squared here.

49
00:02:09,729 --> 00:02:11,680
So the number that should pop out, you say, OK, what two

50
00:02:11,680 --> 00:02:14,730
numbers add up to minus 8 and that when I square

51
00:02:14,729 --> 00:02:16,889
it is equal to 16?

52
00:02:16,889 --> 00:02:17,869
Minus 4, right?

53
00:02:17,870 --> 00:02:20,250
Minus 4, minus 4 is minus 8.

54
00:02:20,250 --> 00:02:22,430
Minus 4 squared is 16.

55
00:02:22,430 --> 00:02:25,110
So the number we're talking about is minus 4b, right?

56
00:02:25,110 --> 00:02:27,910
Minus 4b.

57
00:02:27,909 --> 00:02:29,090
b, that's a b.

58
00:02:29,090 --> 00:02:35,039
Plus minus 4b is equal to minus 8b.

59
00:02:35,039 --> 00:02:41,129
And minus 4b squared is equal to 16b squared.

60
00:02:41,129 --> 00:02:45,310
So we can factor this out to be 3-- let me switch colors.

61
00:02:45,310 --> 00:02:47,390
3 times.

62
00:02:47,389 --> 00:02:48,919
We said minus 4b is the number.

63
00:02:48,919 --> 00:02:50,609
When you add it twice you get minus 8b.

64
00:02:50,610 --> 00:02:51,610
And you square it, you get that.

65
00:02:51,610 --> 00:03:00,900
So it's a minus 4b times a minus 4b.

66
00:03:00,900 --> 00:03:04,230
And that is choice C.

67
00:03:04,229 --> 00:03:08,340
Next problem, 40-- did I skip 45?

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00:03:08,340 --> 00:03:08,800
I did.

69
00:03:08,800 --> 00:03:11,990
Let me copy and paste 45 in here.

70
00:03:11,990 --> 00:03:12,290
Can't

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00:03:12,289 --> 00:03:18,319
skip problems. 45.

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00:03:18,319 --> 00:03:22,150
Copy and paste it in.

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00:03:22,150 --> 00:03:23,099
OK.

74
00:03:23,099 --> 00:03:25,739
Maybe over do on top of this one.

75
00:03:25,740 --> 00:03:28,170
Let me see if I can erase this.

76
00:03:28,169 --> 00:03:29,280
Look at that.

77
00:03:29,280 --> 00:03:33,080
All right, now I can paste 45 and we're ready to go.

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00:03:33,080 --> 00:03:39,670
Which is a factor of x squared minus 11x plus 24?

79
00:03:39,669 --> 00:03:40,709
So there's two ways to do it.

80
00:03:40,710 --> 00:03:42,610
You could just factor this.

81
00:03:42,610 --> 00:03:43,500
Well, that's the easiest way.

82
00:03:43,500 --> 00:03:44,689
Or you could test for 0's.

83
00:03:44,689 --> 00:03:45,560
But we'll just factor.

84
00:03:45,560 --> 00:03:49,750
So if we wanted to say-- well, let me just-- x squared minus

85
00:03:49,750 --> 00:03:52,340
11x plus 24.

86
00:03:52,340 --> 00:03:54,569
So just like the last problem you have to say, what two

87
00:03:54,569 --> 00:03:58,469
numbers when I add them equal to minus 11, and when I

88
00:03:58,469 --> 00:04:02,539
multiply them, is equal to positive 24?

89
00:04:02,539 --> 00:04:03,919
So first of all, if I'm multiplying them and I get a

90
00:04:03,919 --> 00:04:05,889
positive 24, they're either both positive or

91
00:04:05,889 --> 00:04:07,419
they're both negative.

92
00:04:07,419 --> 00:04:09,250
Since this is a negative number it tells me that they

93
00:04:09,250 --> 00:04:11,009
both have-- I mean they both can't be positive.

94
00:04:11,009 --> 00:04:12,819
You can't add two positive numbers and

95
00:04:12,819 --> 00:04:13,889
end up with a negative.

96
00:04:13,889 --> 00:04:16,019
So we're going to deal with two negative numbers.

97
00:04:16,019 --> 00:04:18,170
So essentially, what two negative numbers when I add

98
00:04:18,170 --> 00:04:22,470
them equals minus 11 and when I multiplying them equals 24?

99
00:04:22,470 --> 00:04:25,330
And hopefully, 8 and 3 pop out at you.

100
00:04:25,329 --> 00:04:29,750
Because minus 8 plus minus 3 is equal to minus 11.

101
00:04:29,750 --> 00:04:30,600
That's that.

102
00:04:30,600 --> 00:04:34,300
And then minus 8 times minus 3 is positive 24.

103
00:04:34,300 --> 00:04:41,980
So this can be factored as x minus 8 times x minus 3.

104
00:04:41,980 --> 00:04:43,569
And if we look at their choices, they

105
00:04:43,569 --> 00:04:45,909
had x minus 3 there.

106
00:04:45,910 --> 00:04:48,590
And of course, don't get this confused.

107
00:04:48,589 --> 00:04:51,849
If they said, solve this equation, right?

108
00:04:51,850 --> 00:04:52,900
This is just an expression.

109
00:04:52,899 --> 00:04:55,419
Now it becomes an equation if we put an equal sign.

110
00:04:55,420 --> 00:04:58,620
Then we would have this equaling 0 and then you could

111
00:04:58,620 --> 00:05:00,596
say the roots of this polynomial or the solutions to

112
00:05:00,596 --> 00:05:02,529
the equation are what makes this true.

113
00:05:02,529 --> 00:05:03,750
And then the solutions wouldn't be

114
00:05:03,750 --> 00:05:05,579
minus 8 and minus 3.

115
00:05:05,579 --> 00:05:07,339
The solutions would be what makes this 0.

116
00:05:07,339 --> 00:05:11,269
So it's be x is equal to 8 would make this 0, or x is

117
00:05:11,269 --> 00:05:12,539
equal to 3 would make that 0.

118
00:05:12,540 --> 00:05:14,620
But anyway, I'm going off on a tangent and that's not what

119
00:05:14,620 --> 00:05:15,555
they asked you.

120
00:05:15,555 --> 00:05:18,389
But I think that confuses people sometimes.

121
00:05:18,389 --> 00:05:19,949
OK, 46.

122
00:05:19,949 --> 00:05:24,610
Which of the following shows 9t squared plus 12t plus 4

123
00:05:24,610 --> 00:05:27,620
factored completely?

124
00:05:27,620 --> 00:05:30,410
OK, now this is an interesting one because immediately, when

125
00:05:30,410 --> 00:05:36,980
I look at the numbers, there's not one number that I can just

126
00:05:36,980 --> 00:05:40,340
factor out of everything.

127
00:05:40,339 --> 00:05:43,349
9, 12, and 4, they don't have any common factors, so I can't

128
00:05:43,350 --> 00:05:44,470
just do that simplification.

129
00:05:44,470 --> 00:05:47,880
So we're going to do a little bit more complexity.

130
00:05:47,879 --> 00:05:49,870
But the best way to think about is whatever's on the t

131
00:05:49,870 --> 00:05:59,110
squared, this is probably kind of-- this whole expression, if

132
00:05:59,110 --> 00:06:01,800
we're trying to factor it into two binomials or into one

133
00:06:01,800 --> 00:06:04,870
binomial, this is going to be the first term of that

134
00:06:04,870 --> 00:06:06,310
binomial squared.

135
00:06:06,310 --> 00:06:12,589
So we're going to be dealing with something like 3t.

136
00:06:12,589 --> 00:06:14,879
I'm just taking the square root of 9t squared.

137
00:06:14,879 --> 00:06:17,379
3t plus some number.

138
00:06:17,379 --> 00:06:19,870
Let's say plus a.

139
00:06:19,870 --> 00:06:24,149
Times 3t plus b.

140
00:06:24,149 --> 00:06:26,199
And now we can actually just multiply this out

141
00:06:26,199 --> 00:06:27,519
and see what happens.

142
00:06:27,519 --> 00:06:29,849
Well first of all, they gave us a multiple choice.

143
00:06:29,850 --> 00:06:32,030
We could just multiply these out and see what happens.

144
00:06:32,029 --> 00:06:34,279
But let's pretend like that they didn't give us choices

145
00:06:34,279 --> 00:06:35,809
and we had to do this in a vacuum.

146
00:06:35,810 --> 00:06:37,170
If we did this in a vacuum, we would have

147
00:06:37,170 --> 00:06:37,990
to factor it ourselves.

148
00:06:37,990 --> 00:06:39,340
We wouldn't just be able to test their choices.

149
00:06:39,339 --> 00:06:40,349
Let's do that.

150
00:06:40,350 --> 00:06:42,200
So if we multiply this out, we have 3t

151
00:06:42,199 --> 00:06:46,789
times 3t is 9t squared.

152
00:06:46,790 --> 00:06:50,700
Then you 3t times b, so it's plus 3bt.

153
00:06:50,699 --> 00:06:53,319


154
00:06:53,319 --> 00:06:58,649
Plus 3at plus ab.

155
00:06:58,649 --> 00:07:01,779


156
00:07:01,779 --> 00:07:05,629
So this simplifies to 9t squared plus.

157
00:07:05,629 --> 00:07:06,399
Now what are we adding?

158
00:07:06,399 --> 00:07:06,729
Let's see.

159
00:07:06,730 --> 00:07:09,970
It says 3 times, a plus b, t.

160
00:07:09,970 --> 00:07:16,380
So it'd be 3 times, a plus b, t.

161
00:07:16,379 --> 00:07:19,469
I just added these two terms and I factored out the t and

162
00:07:19,470 --> 00:07:22,410
the 3 plus ab.

163
00:07:22,410 --> 00:07:25,270
Well now we can do a little pattern matching, right?

164
00:07:25,269 --> 00:07:34,839
We could say a plus b times 3 is equal to 12.

165
00:07:34,839 --> 00:07:39,949
So a plus b is equal to what? a plus b is equal to 12.

166
00:07:39,949 --> 00:07:42,610
This whole coefficient right here is a 12 up here.

167
00:07:42,610 --> 00:07:44,879
So a plus b, it must be equal to 4.

168
00:07:44,879 --> 00:07:47,449


169
00:07:47,449 --> 00:07:51,159
Because 4 times 3 is equal to 12. a plus b is equal to 4 and

170
00:07:51,160 --> 00:07:53,095
we have a times b is also equal to 4.

171
00:07:53,095 --> 00:07:55,800


172
00:07:55,800 --> 00:07:57,980
So the only number that I can think of when I add them I get

173
00:07:57,980 --> 00:08:01,650
4, when I multiply them I get 4 is 2 and 2.

174
00:08:01,649 --> 00:08:04,919
Both of these are 2.

175
00:08:04,920 --> 00:08:10,949
So if I were to factor this completely I get 3t plus 2

176
00:08:10,949 --> 00:08:11,920
squared, essentially.

177
00:08:11,920 --> 00:08:13,750
Because both of these terms are the same thing, and

178
00:08:13,750 --> 00:08:15,490
that's choice A.

179
00:08:15,490 --> 00:08:17,840
A faster way frankly, to do this might have just been to

180
00:08:17,839 --> 00:08:20,379
multiply this out and say, that's the same thing.

181
00:08:20,379 --> 00:08:22,209
Anyway, next problem.

182
00:08:22,209 --> 00:08:24,620
What is the complete factorization

183
00:08:24,620 --> 00:08:28,399
of 32 minus 8z squared?

184
00:08:28,399 --> 00:08:30,099
So let's think about this a little bit.

185
00:08:30,100 --> 00:08:33,519
So the first thing that I like to do once again is to try to

186
00:08:33,519 --> 00:08:35,629
factor out any numbers that are just common to all of the

187
00:08:35,629 --> 00:08:37,279
terms. So let me do that.

188
00:08:37,279 --> 00:08:42,470
So 32 minus 8z squared.

189
00:08:42,470 --> 00:08:43,980
8 goes into both of these, right?

190
00:08:43,980 --> 00:08:45,090
So let's factor out an 8.

191
00:08:45,090 --> 00:08:47,649
And actually, let's factor out a negative 8, and I'll show

192
00:08:47,649 --> 00:08:48,230
you why did that.

193
00:08:48,230 --> 00:08:49,639
Because I like to put the z squared, I like

194
00:08:49,639 --> 00:08:50,970
that one to be positive.

195
00:08:50,970 --> 00:08:52,710
So let's factor out a negative 8.

196
00:08:52,710 --> 00:08:56,129
And so you get a minus 8.

197
00:08:56,129 --> 00:08:56,919
And you didn't have to do that.

198
00:08:56,919 --> 00:08:58,149
You could have just factored an 8.

199
00:08:58,149 --> 00:09:00,870
What's 32 divided by minus 8?

200
00:09:00,870 --> 00:09:03,110
That's minus 4.

201
00:09:03,110 --> 00:09:06,730
And then minus 8z squared divided by minus 8 is just

202
00:09:06,730 --> 00:09:09,330
plus z squared.

203
00:09:09,330 --> 00:09:13,009
And so we can rearrange this, so this becomes minus 8, times

204
00:09:13,009 --> 00:09:16,689
z squared minus 4.

205
00:09:16,690 --> 00:09:20,320
And now I'll review this because this is an Algebra 1

206
00:09:20,320 --> 00:09:22,360
test, so maybe this isn't obvious to you.

207
00:09:22,360 --> 00:09:26,870
But in general, if you see something like this, a plus b

208
00:09:26,870 --> 00:09:30,639
times a minus b-- and this might be a good exercise for

209
00:09:30,639 --> 00:09:31,990
you to multiply this out.

210
00:09:31,990 --> 00:09:36,419
But this is equal to a squared minus b squared, because the

211
00:09:36,419 --> 00:09:37,639
middle terms cancel out.

212
00:09:37,639 --> 00:09:39,730
And that's a good exercise for you to do.

213
00:09:39,730 --> 00:09:42,740
So this has that same property.

214
00:09:42,740 --> 00:09:44,930
This is a squared minus b squared.

215
00:09:44,929 --> 00:09:46,449
This is a perfect square.

216
00:09:46,450 --> 00:09:51,720
So then this up here can be factored as minus 8 times--

217
00:09:51,720 --> 00:09:55,350
this is a squared minus b squared.

218
00:09:55,350 --> 00:10:01,860
So if we could say a is z and b is 2.

219
00:10:01,860 --> 00:10:03,940
Because then when we would have z squared minus 2

220
00:10:03,940 --> 00:10:05,520
squared, which is 4.

221
00:10:05,519 --> 00:10:15,179
So if a is z, so it's z plus 2 times z minus 2.

222
00:10:15,179 --> 00:10:17,250
This is probably the most common thing you'll see in a

223
00:10:17,250 --> 00:10:18,720
lot of factoring examples.

224
00:10:18,720 --> 00:10:20,620
You'll see something that has this pattern, a squared minus

225
00:10:20,620 --> 00:10:22,409
b squared, and you should easily be able

226
00:10:22,409 --> 00:10:23,120
to recognize it.

227
00:10:23,120 --> 00:10:25,659
You should be able to prove it to yourself as well that this

228
00:10:25,659 --> 00:10:28,850
is the same thing as a plus b times a minus b.

229
00:10:28,850 --> 00:10:33,250
Or in this case, z plus 2 times z minus 2.

230
00:10:33,250 --> 00:10:36,629
And that is z minus 2.

231
00:10:36,629 --> 00:10:39,039
So this is interesting.

232
00:10:39,039 --> 00:10:41,899
So I don't see exactly what we wrote.

233
00:10:41,899 --> 00:10:44,829
I have a minus 8, but we want a z plus 2 and z minus 2.

234
00:10:44,830 --> 00:10:46,139
But they don't have that.

235
00:10:46,139 --> 00:10:50,399
But what they do have is this-- let me rewrite it.

236
00:10:50,399 --> 00:10:52,750
They don't have any of it, but maybe they have-- they want

237
00:10:52,750 --> 00:10:55,830
the minus to be multiplied by one of these other terms. So

238
00:10:55,830 --> 00:10:58,460
what we could do is we could multiply the minus because

239
00:10:58,460 --> 00:10:59,820
multiplication is associative.

240
00:10:59,820 --> 00:11:01,270
It doesn't matter what order you do it in.

241
00:11:01,269 --> 00:11:08,269
So we could rewrite this as 8 times z plus 2 times minus 1

242
00:11:08,269 --> 00:11:11,639
times z minus 2.

243
00:11:11,639 --> 00:11:15,480
And then this becomes 8 times z plus 2.

244
00:11:15,480 --> 00:11:17,960
This isn't minus 1, this is times minus 1.

245
00:11:17,960 --> 00:11:19,840
Times what?

246
00:11:19,840 --> 00:11:22,820
Minus z plus 2.

247
00:11:22,820 --> 00:11:25,900
And now I think that choice is there.

248
00:11:25,899 --> 00:11:27,350
Right.

249
00:11:27,350 --> 00:11:31,790
So we can rewrite this as 8 times 2 plus z-- I'm just

250
00:11:31,789 --> 00:11:34,980
rearranging it-- times 2 minus z.

251
00:11:34,980 --> 00:11:38,409
And then that is choice B.

252
00:11:38,409 --> 00:11:40,980
And so really, given the choices they gave us, it

253
00:11:40,980 --> 00:11:43,820
probably would have been faster to just factor out an 8

254
00:11:43,820 --> 00:11:45,600
from the beginning and not a positive 8.

255
00:11:45,600 --> 00:11:47,800
And you would have immediately gotten to 2 plus z

256
00:11:47,799 --> 00:11:49,139
times 2 minus z.

257
00:11:49,139 --> 00:11:52,009
Anyway, I'll see you in the next video.