WEBVTT
1
00:00:00.000 --> 00:00:07.600
Multiplying polynomials part two.
2
00:00:07.600 --> 00:00:14.600
We're going to go over several methods for multiplying a polynomial by a polynomial.
3
00:00:14.600 --> 00:00:19.940
I'm going to start with a little reminder of the distributive property, a times b plus
4
00:00:19.940 --> 00:00:26.640
c equals a times b plus a times c.
5
00:00:26.640 --> 00:00:36.880
Now, what if r a was actually something like 3x plus 5 to make a little bit more complicated?
6
00:00:36.880 --> 00:00:43.920
And then we had that times 4x minus 7.
7
00:00:43.920 --> 00:00:52.280
In other words, you kind of think of this whole thing as an a and a b and there's your
8
00:00:52.280 --> 00:00:55.080
like a minusc in this case.
9
00:00:55.080 --> 00:00:59.800
By using the distributive property, it would mean you would take this whole first part,
10
00:00:59.800 --> 00:01:13.480
your a, 3x plus 5 times a 4x minus that whole 3x plus 5 times a 7.
11
00:01:13.480 --> 00:01:17.040
That's kind of complicated looking, isn't it?
12
00:01:17.040 --> 00:01:23.240
And in fact, I don't like doing it this way, but I'm going to show you where other methods
13
00:01:23.240 --> 00:01:24.240
come from.
14
00:01:24.240 --> 00:01:27.640
Now, it's easier when you have the 4x in front of the parentheses, right?
15
00:01:27.640 --> 00:01:32.240
So let's just rewrite that as 4x times 3x plus 5.
16
00:01:32.240 --> 00:01:36.280
And let's put the 7 in front of the 3x plus 5 because of the commutative property, it
17
00:01:36.280 --> 00:01:39.560
doesn't matter the order when you multiply, correct?
18
00:01:39.560 --> 00:01:40.960
So I could put the 4x in front.
19
00:01:40.960 --> 00:01:43.320
And that's easier to do the distributive property here.
20
00:01:43.320 --> 00:01:45.560
So we have 4x times 3x plus 5.
21
00:01:45.560 --> 00:01:50.360
Now, I just have the easier distributive property.
22
00:01:50.360 --> 00:01:55.080
4x times 3x is 12x squared.
23
00:01:55.080 --> 00:01:59.880
4x times 5 is plus 20x.
24
00:01:59.880 --> 00:02:05.000
Negative 7 times 3x, be careful of your signs, is negative 21x.
25
00:02:05.000 --> 00:02:09.720
And negative 7 times 5 is minus 35.
26
00:02:09.720 --> 00:02:14.880
I still have to add like terms.
27
00:02:14.880 --> 00:02:22.040
So we have 12x squared, and here are two light terms, 20x minus 21x.
28
00:02:22.040 --> 00:02:32.320
That's a negative 1x, which we just write as minus x minus 35.
29
00:02:32.320 --> 00:02:42.000
So that's one way of multiplying 3x plus 5 times 4x minus 7.
30
00:02:42.000 --> 00:02:43.800
Kind of a long way to do it, though.
31
00:02:43.800 --> 00:02:48.760
So I'm going to show you some easier ways of getting exactly the same answer.
32
00:02:48.760 --> 00:02:52.360
So let's take this problem again.
33
00:02:52.360 --> 00:02:59.680
3x plus 5 times 4x minus 7.
34
00:02:59.680 --> 00:03:02.320
And before going on, notice what happened.
35
00:03:02.320 --> 00:03:05.320
I've got some multiplications here.
36
00:03:05.320 --> 00:03:09.720
I have a 4x times 3x, and I have a 4x times 5.
37
00:03:09.720 --> 00:03:15.680
In other words, this 4x, up in the original problem, got multiplied by each thing in this
38
00:03:15.680 --> 00:03:19.320
parentheses by the 3x and the 5.
39
00:03:19.320 --> 00:03:26.120
And then I had a negative 7, this number in the second parentheses, was multiplied by
40
00:03:26.120 --> 00:03:30.880
each thing in the first parentheses, 3x plus 5, as well.
41
00:03:30.880 --> 00:03:36.440
So another way of doing this is a vertical format.
42
00:03:36.440 --> 00:03:43.800
We could write the 3x plus 5 and the 4x minus 7.
43
00:03:43.800 --> 00:03:47.840
You just write one of the polynomials on top of the other one.
44
00:03:47.840 --> 00:03:51.400
And if it's three terms, you would just score a lot of it further.
45
00:03:51.400 --> 00:03:55.360
And what we're going to do is multiply the 4x times everything in the top and then the
46
00:03:55.360 --> 00:03:57.400
negative 7 times everything in the top.
47
00:03:57.400 --> 00:03:59.400
So let's do that.
48
00:03:59.400 --> 00:04:07.240
4x times 3x is 12x squared, and then the 4x times the 5 is plus 20x.
49
00:04:07.240 --> 00:04:10.160
All right, so we've just multiplied 4x times.
50
00:04:10.160 --> 00:04:14.400
Remember, the 4x got multiplied by this whole thing?
51
00:04:14.400 --> 00:04:18.160
And now we're going to do the negative 7 times 3x plus 5.
52
00:04:18.160 --> 00:04:19.800
So I skew it over.
53
00:04:19.800 --> 00:04:28.680
Negative 7 times 3x is negative 21x, and negative 7 times 5 is negative 35.
54
00:04:28.680 --> 00:04:36.600
These are the same 4 terms I got right there.
55
00:04:36.600 --> 00:04:44.000
12x squared plus 20x minus 21x minus 35.
56
00:04:44.000 --> 00:04:48.040
12x squared plus 20x minus 21x minus 35.
57
00:04:48.040 --> 00:04:49.680
The exact same 4 terms.
58
00:04:49.680 --> 00:04:53.480
It's a little easier to keep track of here.
59
00:04:53.480 --> 00:05:00.960
And now I need to add the like term, so the 12x squared, 20 minus 21 is minus 1, which
60
00:05:00.960 --> 00:05:09.160
we write as minus x, and minus 35.
61
00:05:09.160 --> 00:05:15.800
So this is a vertical format, which most people find a lot easier to do than the first
62
00:05:15.800 --> 00:05:18.960
way with the distributive property.
63
00:05:18.960 --> 00:05:23.560
Okay, so let's have you try one now.
64
00:05:23.560 --> 00:05:31.040
All right, so put the video on pause and multiply x minus 8 times 2x plus 1, either using
65
00:05:31.040 --> 00:05:35.520
the distributive property the first way I showed, or using a vertical format.
66
00:05:35.520 --> 00:05:39.280
And then when you're done, come back.
67
00:05:39.280 --> 00:05:46.360
Okay, I like to be the vertical format, so I'm going to write x minus 8 times 2x plus
68
00:05:46.360 --> 00:05:56.880
1, and I have 2x times x is 2x squared, and 2x times negative 8 is negative 16x.
69
00:05:56.880 --> 00:06:00.840
So I've now just multiplied the 2x times the x minus 8.
70
00:06:00.840 --> 00:06:04.560
Now I've got to do the 1 times x minus 8.
71
00:06:04.560 --> 00:06:07.320
One times x is plus x.
72
00:06:07.320 --> 00:06:13.400
Make sure you light it up under its like term, and you could put just plus x or plus 1x.
73
00:06:13.400 --> 00:06:18.240
And one times negative 8 is minus 8.
74
00:06:18.240 --> 00:06:19.920
So we've got 2x squared.
75
00:06:19.920 --> 00:06:22.520
I'm going to add the 2 light terms here.
76
00:06:22.520 --> 00:06:29.120
That's a negative 15x minus 8.
77
00:06:29.120 --> 00:06:30.840
And there's our answer.
78
00:06:30.840 --> 00:06:35.440
Keep in mind, the multiplication is going on right here.
79
00:06:35.440 --> 00:06:40.440
This is the 2x times x minus 8.
80
00:06:40.440 --> 00:06:44.920
And this is the 1 times x minus 8.
81
00:06:44.920 --> 00:06:48.120
So see what's happening here?
82
00:06:48.120 --> 00:06:52.520
I'm doing the 2x times the x minus 8, and I'm doing the 1.
83
00:06:52.520 --> 00:06:58.720
So there's the distributive property how it works again.
84
00:06:58.720 --> 00:07:08.720
All right, let's do a problem that has 3 times 2 times 2.
85
00:07:08.720 --> 00:07:15.040
All right, so let's multiply 2x squared minus 5x plus 1 times 3x minus 4.
86
00:07:15.040 --> 00:07:17.040
I'm going to use a vertical format.
87
00:07:17.040 --> 00:07:21.400
It's always easiest to write the one that has the most terms in it in the top, but you
88
00:07:21.400 --> 00:07:23.200
don't have to.
89
00:07:23.200 --> 00:07:27.440
So we're going to first do 3x times every term in the top.
90
00:07:27.440 --> 00:07:32.480
So 3x times 2x squared, 6x cubed, right?
91
00:07:32.480 --> 00:07:34.640
X squared times x is x cubed.
92
00:07:34.640 --> 00:07:43.800
So 3x times negative 5x is negative 15x squared, and 3x times 1 is plus 3x.
93
00:07:43.800 --> 00:07:47.480
So we've just multiplied the 3x times everything here, right?
94
00:07:47.480 --> 00:07:50.880
We've done the 3x times everything there.
95
00:07:50.880 --> 00:07:55.200
And now we're going to do negative 4 times everything in the top.
96
00:07:55.200 --> 00:08:00.560
Negative 4 times 2x squared is negative 8x squared.
97
00:08:00.560 --> 00:08:07.680
Negative 4 times 5x is negative 20x, and negative 4 times 1 is negative 4.
98
00:08:07.680 --> 00:08:10.200
So I'm being careful that I do line up my light terms.
99
00:08:10.200 --> 00:08:15.760
If you write the exponents in descending order like the x squared term, the x in the constant,
100
00:08:15.760 --> 00:08:23.200
the x and then the constant, usually if you move over like this, it'll line up nicely.
101
00:08:23.200 --> 00:08:26.960
So now we're going to add light terms, 6x cubed.
102
00:08:26.960 --> 00:08:34.160
Now be careful here, it's a negative 15 plus a negative 8, that's a negative 23x squared.
103
00:08:34.160 --> 00:08:41.000
And a 3 plus negative 20 is negative 17x, and then minus 4.
104
00:08:41.000 --> 00:08:44.640
So here we go.
105
00:08:44.640 --> 00:08:48.200
We just multiplied 2 polynomials where 1 was a trinomial.
106
00:08:48.200 --> 00:08:52.480
It had 3 terms, and the other one was a minomial.
107
00:08:52.480 --> 00:09:01.240
All right, I want you to put the video on pause and multiply x squared minus 3x plus 2 times
108
00:09:01.240 --> 00:09:04.720
x squared minus 4x minus 6.
109
00:09:04.720 --> 00:09:10.320
This time you've got 3 terms, times 3 other terms, you'll end up with 9 multiplications.
110
00:09:10.320 --> 00:09:13.120
Are you ready?
111
00:09:13.120 --> 00:09:16.800
x squared times x squared is x to the 4th.
112
00:09:16.800 --> 00:09:27.800
x squared times minus 3x is minus 3x cubed, and x squared times 2 is plus 2x squared.
113
00:09:27.800 --> 00:09:28.800
Moving over.
114
00:09:28.800 --> 00:09:34.440
Now we have negative 4x times x squared is negative 4x cubed.
115
00:09:34.440 --> 00:09:40.080
Negative 4x times negative 3x is plus 12x squared.
116
00:09:40.080 --> 00:09:44.640
Negative 4x plus 2 is minus 8x.
117
00:09:44.640 --> 00:09:52.280
This one, negative 6 times 6x squared is negative 6x squared.
118
00:09:52.280 --> 00:09:59.960
Negative 6 times negative 3x is plus 18x, and negative 6 times 2 is negative 12.
119
00:09:59.960 --> 00:10:03.200
See how I got 9 different terms?
120
00:10:03.200 --> 00:10:07.280
Now I just add the like terms, and I will be done.
121
00:10:07.280 --> 00:10:11.040
And this is just one method.
122
00:10:11.040 --> 00:10:17.680
And go to the next video, and I'll show you another way to get the same answer.