0:00:00.000,0:00:00.360


0:00:00.360,0:00:03.870
In this video I want to do a[br]bunch of examples of factoring

0:00:03.870,0:00:06.590
a second degree polynomial,[br]which is

0:00:06.590,0:00:08.870
often called a quadratic.

0:00:08.870,0:00:12.650
Sometimes a quadratic[br]polynomial, or just a

0:00:12.650,0:00:15.850
quadratic itself, or quadratic[br]expression, but all it means

0:00:15.850,0:00:18.300
is a second degree polynomial.

0:00:18.300,0:00:22.440
So something that's going to[br]have a variable raised to the

0:00:22.440,0:00:23.080
second power.

0:00:23.080,0:00:26.460
In this case, in all of the[br]examples we'll do, it'll be x.

0:00:26.460,0:00:30.740
So let's say I have the[br]quadratic expression, x

0:00:30.740,0:00:35.280
squared plus 10x, plus 9.

0:00:35.280,0:00:39.870
And I want to factor it into the[br]product of two binomials.

0:00:39.870,0:00:41.520
How do we do that?

0:00:41.520,0:00:44.440
Well, let's just think about[br]what happens if we were to

0:00:44.440,0:00:51.690
take x plus a, and multiply[br]that by x plus b.

0:00:51.690,0:00:55.050
If we were to multiply these[br]two things, what happens?

0:00:55.050,0:00:57.090
Well, we have a little bit[br]of experience doing this.

0:00:57.090,0:01:03.170
This will be x times x, which is[br]x squared, plus x times b,

0:01:03.170,0:01:12.690
which is bx, plus a times x,[br]plus a times b-- plus ab.

0:01:12.690,0:01:15.800
Or if we want to add these two[br]in the middle right here,

0:01:15.800,0:01:18.890
because they're both[br]coefficients of x.

0:01:18.890,0:01:22.102
We could right this as x squared[br]plus-- I can write it

0:01:22.102,0:01:29.700
as b plus a, or a plus[br]b, x, plus ab.

0:01:29.700,0:01:34.390
So in general, if we assume that[br]this is the product of

0:01:34.390,0:01:40.765
two binomials, we see that this[br]middle coefficient on the

0:01:40.765,0:01:43.250
x term, or you could say the[br]first degree coefficient

0:01:43.250,0:01:49.040
there, that's going to be[br]the sum of our a and b.

0:01:49.040,0:01:51.360
And then the constant term is[br]going to be the product

0:01:51.360,0:01:52.520
of our a and b.

0:01:52.520,0:01:57.290
Notice, this would map[br]to this, and this

0:01:57.290,0:01:58.790
would map to this.

0:01:58.790,0:02:02.600
And, of course, this is the[br]same thing as this.

0:02:02.600,0:02:05.590
So can we somehow pattern[br]match this to that?

0:02:05.590,0:02:14.060
Is there some a and b where[br]a plus b is equal to 10?

0:02:14.060,0:02:22.070
And a times b is equal to 9?

0:02:22.070,0:02:23.850
Well, let's just think about[br]it a little bit.

0:02:23.850,0:02:25.470
What are the factors of 9?

0:02:25.470,0:02:27.770
What are the things that a[br]and b could be equal to?

0:02:27.770,0:02:29.170
And we're assuming that[br]everything is an integer.

0:02:29.170,0:02:32.350
And normally when we're[br]factoring, especially when

0:02:32.350,0:02:33.950
we're beginning to factor,[br]we're dealing

0:02:33.950,0:02:35.580
with integer numbers.

0:02:35.580,0:02:37.080
So what are the factors of 9?

0:02:37.080,0:02:40.730
They're 1, 3, and 9.

0:02:40.730,0:02:45.000
So this could be a 3 and a 3,[br]or it could be a 1 and a 9.

0:02:45.000,0:02:48.630
Now, if it's a 3 and a 3, then[br]you'll have 3 plus 3-- that

0:02:48.630,0:02:49.840
doesn't equal 10.

0:02:49.840,0:02:53.760
But if it's a 1 and a[br]9, 1 times 9 is 9.

0:02:53.760,0:02:56.670
1 plus 9 is 10.

0:02:56.670,0:02:57.570
So it does work.

0:02:57.570,0:03:04.190
So a could be equal to 1, and[br]b could be equal to 9.

0:03:04.190,0:03:08.920
So we could factor this[br]as being x plus 1,

0:03:08.920,0:03:12.970
times x plus 9.

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And if you multiply these two[br]out, using the skills we

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developed in the last few[br]videos, you'll see that it is

0:03:18.970,0:03:22.880
indeed x squared plus[br]10x, plus 9.

0:03:22.880,0:03:25.150
So when you see something like[br]this, when the coefficient on

0:03:25.150,0:03:28.070
the x squared term, or the[br]leading coefficient on this

0:03:28.070,0:03:31.650
quadratic is a 1, you can just[br]say, all right, what two

0:03:31.650,0:03:34.530
numbers add up to this[br]coefficient right here?

0:03:34.530,0:03:37.720


0:03:37.720,0:03:39.870
And those same two numbers, when[br]you take their product,

0:03:39.870,0:03:41.660
have to be equal to 9.

0:03:41.660,0:03:43.510
And of course, this has to[br]be in standard form.

0:03:43.510,0:03:46.000
Or if it's not in standard form,[br]you should put it in

0:03:46.000,0:03:48.500
that form, so that you can[br]always say, OK, whatever's on

0:03:48.500,0:03:51.530
the first degree coefficient,[br]my a and b

0:03:51.530,0:03:52.300
have to add to that.

0:03:52.300,0:03:55.890
Whatever's my constant term, my[br]a times b, the product has

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to be that.

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Let's do several[br]more examples.

0:03:58.150,0:04:00.510
I think the more examples[br]we do the more

0:04:00.510,0:04:02.630
sense this'll make.

0:04:02.630,0:04:08.700
Let's say we had x squared[br]plus 10x, plus-- well, I

0:04:08.700,0:04:11.100
already did 10x, let's do a[br]different number-- x squared

0:04:11.100,0:04:15.366
plus 15x, plus 50.

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And we want to factor this.

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Well, same drill.

0:04:20.339,0:04:22.600
We have an x squared term.

0:04:22.600,0:04:25.130
We have a first degree term.

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This right here should be[br]the sum of two numbers.

0:04:27.960,0:04:30.620
And then this term, the constant[br]term right here,

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should be the product[br]of two numbers.

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So we need to think of two[br]numbers that, when I multiply

0:04:35.640,0:04:39.220
them I get 50, and when[br]I add them, I get 15.

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And this is going to be a bit of[br]an art that you're going to

0:04:41.910,0:04:44.530
develop, but the more practice[br]you do, you're going to see

0:04:44.530,0:04:45.730
that it'll start to[br]come naturally.

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So what could a and b be?

0:04:47.330,0:04:48.975
Let's think about the[br]factors of 50.

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It could be 1 times 50.

0:04:52.230,0:04:55.072
2 times 25.

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Let's see, 4 doesn't[br]go into 50.

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It could be 5 times 10.

0:05:02.450,0:05:03.580
I think that's all of them.

0:05:03.580,0:05:05.920
Let's try out these numbers,[br]and see if any of

0:05:05.920,0:05:07.310
these add up to 15.

0:05:07.310,0:05:12.630
So 1 plus 50 does not[br]add up to 15.

0:05:12.630,0:05:16.160
2 plus 25 does not[br]add up to 15.

0:05:16.160,0:05:19.260
But 5 plus 10 does[br]add up to 15.

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So this could be 5 plus 10, and[br]this could be 5 times 10.

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So if we were to factor this,[br]this would be equal to x plus

0:05:28.760,0:05:32.630
5, times x plus 10.

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And multiply it out.

0:05:33.840,0:05:36.710
I encourage you to multiply this[br]out, and see that this is

0:05:36.710,0:05:39.660
indeed x squared plus[br]15x, plus 10.

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In fact, let's do it. x[br]times x, x squared. x

0:05:43.320,0:05:45.800
times 10, plus 10x.

0:05:45.800,0:05:48.600
5 times x, plus 5x.

0:05:48.600,0:05:51.620
5 times 10, plus 50.

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Notice, the 5 times[br]10 gave us the 50.

0:05:55.220,0:06:00.890
The 5x plus the 10x is giving[br]us the 15x in between.

0:06:00.890,0:06:06.560
So it's x squared plus[br]15x, plus 50.

0:06:06.560,0:06:09.440
Let's up the stakes a little[br]bit, introduce some negative

0:06:09.440,0:06:11.040
signs in here.

0:06:11.040,0:06:18.890
Let's say I had x squared[br]minus 11x, plus 24.

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Now, it's the exact[br]same principle.

0:06:21.610,0:06:24.580
I need to think of two numbers,[br]that when I add them,

0:06:24.580,0:06:26.570
need to be equal[br]to negative 11.

0:06:26.570,0:06:30.150
a plus b need to be equal[br]to negative 11.

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And a times b need to[br]be equal to 24.

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Now, there's something for[br]you to think about.

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When I multiply both of these[br]numbers, I'm getting a

0:06:43.960,0:06:45.080
positive number.

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I'm getting a 24.

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That means that both of these[br]need to be positive, or both

0:06:50.260,0:06:51.380
of these need to be negative.

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That's the only way I'm going to[br]get a positive number here.

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Now, if when I add them, I get[br]a negative number, if these

0:06:58.290,0:07:00.720
were positive, there's no way I[br]can add two positive numbers

0:07:00.720,0:07:03.280
and get a negative number, so[br]the fact that their sum is

0:07:03.280,0:07:05.750
negative, and the fact that[br]their product is positive,

0:07:05.750,0:07:10.420
tells me that both a[br]and b are negative.

0:07:10.420,0:07:13.200
a and b have to be negative.

0:07:13.200,0:07:15.730
Remember, one can't be negative[br]and the other one

0:07:15.730,0:07:18.710
can't be positive, because the[br]product would be negative.

0:07:18.710,0:07:22.687
And they both can't be positive,[br]because when you add

0:07:22.687,0:07:24.520
them it would get you[br]a positive number.

0:07:24.520,0:07:27.500
So let's just think about[br]what a and b can be.

0:07:27.500,0:07:28.990
So two negative numbers.

0:07:28.990,0:07:31.260
So let's think about[br]the factors of 24.

0:07:31.260,0:07:33.140
And we'll kind of have to think[br]of the negative factors.

0:07:33.140,0:07:44.670
But let me see, it could be 1[br]times 24, 2 times 11, 3 times

0:07:44.670,0:07:48.070
8, or 4 times 6.

0:07:48.070,0:07:51.220
Now, which of these when I[br]multiply these-- well,

0:07:51.220,0:07:54.380
obviously when I multiply[br]1 times 24, I get 24.

0:07:54.380,0:07:58.910
When I get 2 times 11-- sorry,[br]this is 2 times 12.

0:07:58.910,0:07:59.790
I get 24.

0:07:59.790,0:08:03.090
So we know that all these,[br]the products are 24.

0:08:03.090,0:08:07.470
But which two of these, which[br]two factors, when I add them,

0:08:07.470,0:08:08.790
should I get 11?

0:08:08.790,0:08:09.880
And then we could say,[br]let's take the

0:08:09.880,0:08:11.450
negative of both of those.

0:08:11.450,0:08:15.470
So when you look at these,[br]3 and 8 jump out.

0:08:15.470,0:08:19.150
3 times 8 is equal to 24.

0:08:19.150,0:08:22.810
3 plus 8 is equal to 11.

0:08:22.810,0:08:24.680
But that doesn't quite[br]work out, right?

0:08:24.680,0:08:26.510
Because we have a negative[br]11 here.

0:08:26.510,0:08:29.690
But what if we did negative[br]3 and negative 8?

0:08:29.690,0:08:37.789
Negative 3 times negative 8[br]is equal to positive 24.

0:08:37.789,0:08:43.580
Negative 3 plus negative 8[br]is equal to negative 11.

0:08:43.580,0:08:46.600
So negative 3 and[br]negative 8 work.

0:08:46.600,0:08:53.840
So if we factor this, x squared[br]minus 11x, plus 24 is

0:08:53.840,0:09:02.940
going to be equal to x minus[br]3, times x minus 8.

0:09:02.940,0:09:06.270
Let's do another[br]one like that.

0:09:06.270,0:09:08.330
Actually, let's mix it[br]up a little bit.

0:09:08.330,0:09:19.810
Let's say I had x squared[br]plus 5x, minus 14.

0:09:19.810,0:09:21.770
So here we have a different[br]situation.

0:09:21.770,0:09:26.460
The product of my two numbers is[br]negative, right? a times b

0:09:26.460,0:09:28.190
is equal to negative 14.

0:09:28.190,0:09:29.920
My product is negative.

0:09:29.920,0:09:32.930
That tells me that one of them[br]is positive, and one of them

0:09:32.930,0:09:33.910
is negative.

0:09:33.910,0:09:39.200
And when I add the two, a plus[br]b, it'd be equal to 5.

0:09:39.200,0:09:41.360
So let's think about[br]the factors of 14.

0:09:41.360,0:09:44.300
And what combinations of them,[br]when I add them, if one is

0:09:44.300,0:09:46.560
positive and one is negative,[br]or I'm really kind of taking

0:09:46.560,0:09:49.780
the difference of the[br]two, do I get 5?

0:09:49.780,0:09:53.450
So if I take 1 and 14-- I'm just[br]going to try out things--

0:09:53.450,0:10:01.820
1 and 14, negative 1 plus[br]14 is negative 13.

0:10:01.820,0:10:04.260
Negative 1 plus 14 is 13.

0:10:04.260,0:10:07.430
So let me write all of the[br]combinations that I could do.

0:10:07.430,0:10:09.440
And eventually your brain[br]will just zone in on it.

0:10:09.440,0:10:16.490
So you've got negative 1[br]plus 14 is equal to 13.

0:10:16.490,0:10:20.460
And 1 plus negative 14 is[br]equal to negative 13.

0:10:20.460,0:10:21.380
So those don't work.

0:10:21.380,0:10:22.950
That doesn't equal 5.

0:10:22.950,0:10:24.860
Now what about 2 and 7?

0:10:24.860,0:10:29.600
If I do negative 2-- let me do[br]this in a different color-- if

0:10:29.600,0:10:35.290
I do negative 2 plus 7,[br]that is equal to 5.

0:10:35.290,0:10:35.750
We're done!

0:10:35.750,0:10:36.670
That worked!

0:10:36.670,0:10:39.440
I mean, we could have tried 2[br]plus negative 7, but that'd be

0:10:39.440,0:10:41.070
equal to negative 5, so that[br]wouldn't have worked.

0:10:41.070,0:10:42.960
But negative 2 plus 7 works.

0:10:42.960,0:10:46.590
And negative 2 times[br]7 is negative 14.

0:10:46.590,0:10:47.600
So there we have it.

0:10:47.600,0:10:53.210
We know it's x minus[br]2, times x plus 7.

0:10:53.210,0:10:54.330
That's pretty neat.

0:10:54.330,0:10:56.950
Negative 2 times 7[br]is negative 14.

0:10:56.950,0:11:00.880
Negative 2 plus 7[br]is positive 5.

0:11:00.880,0:11:03.760


0:11:03.760,0:11:07.680
Let's do several more of these,[br]just to really get well

0:11:07.680,0:11:09.520
honed this skill.

0:11:09.520,0:11:16.360
So let's say we have x squared[br]minus x, minus 56.

0:11:16.360,0:11:19.570
So the product of the two[br]numbers have to be minus 56,

0:11:19.570,0:11:21.620
have to be negative 56.

0:11:21.620,0:11:24.430
And their difference, because[br]one is going to be positive,

0:11:24.430,0:11:26.280
and one is going to be[br]negative, right?

0:11:26.280,0:11:28.350
Their difference has[br]to be negative 1.

0:11:28.350,0:11:30.130
And the numbers that immediately[br]jump out in my

0:11:30.130,0:11:31.900
brain-- and I don't know if they[br]jump out in your brain,

0:11:31.900,0:11:33.740
we just learned this in[br]the times tables--

0:11:33.740,0:11:36.480
56 is 8 times 7.

0:11:36.480,0:11:37.480
I mean, there's other numbers.

0:11:37.480,0:11:39.950
It's also 28 times 2.

0:11:39.950,0:11:41.140
It's all sorts of things.

0:11:41.140,0:11:44.300
But 8 times 7 really jumped[br]out into my brain, because

0:11:44.300,0:11:45.470
they're very close[br]to each other.

0:11:45.470,0:11:47.730
And we need numbers that are[br]very close to each other.

0:11:47.730,0:11:50.430
And one of these has to be[br]positive, and one of these has

0:11:50.430,0:11:51.810
to be negative.

0:11:51.810,0:11:55.250
Now, the fact that when their[br]sum is negative, tells me that

0:11:55.250,0:11:58.460
the larger of these two should[br]probably be negative.

0:11:58.460,0:12:01.350
So if we take negative[br]8 times 7, that's

0:12:01.350,0:12:03.320
equal to negative 56.

0:12:03.320,0:12:08.470
And then if we take negative[br]8 plus 7, that is equal to

0:12:08.470,0:12:12.100
negative 1, which is exactly the[br]coefficient right there.

0:12:12.100,0:12:16.490
So when I factor this, this[br]is going to be x minus 8,

0:12:16.490,0:12:18.690
times x plus 7.

0:12:18.690,0:12:21.695
This is often one of the hardest[br]concepts people learn

0:12:21.695,0:12:23.920
in algebra, because it[br]is a bit of an art.

0:12:23.920,0:12:26.710
You have to look at all of the[br]factors here, play with the

0:12:26.710,0:12:29.710
positive and negative signs,[br]see which of those factors

0:12:29.710,0:12:31.900
when one is positive, one is[br]negative, add up to the

0:12:31.900,0:12:33.590
coefficient on the x term.

0:12:33.590,0:12:35.860
But as you do more and more[br]practice, you'll see that

0:12:35.860,0:12:39.280
it'll become a bit[br]of second nature.

0:12:39.280,0:12:42.350
Now let's step up the stakes[br]a little bit more.

0:12:42.350,0:12:46.140
Let's say we had negative x[br]squared-- everything we've

0:12:46.140,0:12:49.040
done so far had a positive[br]coefficient, a positive 1

0:12:49.040,0:12:50.690
coefficient on the[br]x squared term.

0:12:50.690,0:12:55.590
But let's say we had a[br]negative x squared

0:12:55.590,0:12:59.440
minus 5x, plus 24.

0:12:59.440,0:13:00.910
How do we do this?

0:13:00.910,0:13:03.420
Well, the easiest way I can[br]think of doing it is factor

0:13:03.420,0:13:05.670
out a negative 1, and then it[br]becomes just like the problems

0:13:05.670,0:13:07.260
we've been doing before.

0:13:07.260,0:13:11.660
So this is the same thing as[br]negative 1 times positive x

0:13:11.660,0:13:15.990
squared, plus 5x, minus 24.

0:13:15.990,0:13:16.300
Right?

0:13:16.300,0:13:18.080
I just factored a[br]negative 1 out.

0:13:18.080,0:13:20.180
You can multiply negative 1[br]times all of these, and you'll

0:13:20.180,0:13:21.690
see it becomes this.

0:13:21.690,0:13:23.555
Or you could factor the negative[br]1 out and divide all

0:13:23.555,0:13:25.000
of these by negative 1.

0:13:25.000,0:13:26.760
And you get that right there.

0:13:26.760,0:13:29.360
Now, same game as before.

0:13:29.360,0:13:33.510
I need two numbers, that when[br]I take their product I get

0:13:33.510,0:13:34.690
negative 24.

0:13:34.690,0:13:37.125
So one will be positive,[br]one will be negative.

0:13:37.125,0:13:41.550


0:13:41.550,0:13:43.770
When I take their sum,[br]it's going to be 5.

0:13:43.770,0:13:48.850
So let's think about[br]24 is 1 and 24.

0:13:48.850,0:13:55.750
Let's see, if this is negative 1[br]and 24, it'd be positive 23,

0:13:55.750,0:13:57.540
if it was the other way around,[br]it'd be negative 23.

0:13:57.540,0:13:58.500
Doesn't work.

0:13:58.500,0:14:01.210
What about 2 and 12?

0:14:01.210,0:14:04.530
Well, if this is negative--[br]remember, one of these has to

0:14:04.530,0:14:05.040
be negative.

0:14:05.040,0:14:07.570
If the 2 is negative, their[br]sum would be 10.

0:14:07.570,0:14:09.750
If the 12 is negative, their[br]sum would be negative 10.

0:14:09.750,0:14:11.300
Still doesn't work.

0:14:11.300,0:14:13.290
3 and 8.

0:14:13.290,0:14:16.850
If the 3 is negative,[br]their sum will be 5.

0:14:16.850,0:14:17.910
So it works!

0:14:17.910,0:14:24.620
So if we pick negative 3 and[br]8, negative 3 and 8 work.

0:14:24.620,0:14:26.630
Because negative[br]3 plus 8 is 5.

0:14:26.630,0:14:29.510
Negative 3 times 8[br]is negative 24.

0:14:29.510,0:14:31.640
So this is going to be equal[br]to-- can't forget that

0:14:31.640,0:14:35.230
negative 1 out front, and then[br]we factor the inside.

0:14:35.230,0:14:42.990
Negative 1 times x minus[br]3, times x plus 8.

0:14:42.990,0:14:44.800
And if you really wanted to,[br]you could multiply the

0:14:44.800,0:14:46.450
negative 1 times this,[br]you would get 3

0:14:46.450,0:14:47.560
minus x if you did.

0:14:47.560,0:14:48.810
Or you don't have to.

0:14:48.810,0:14:51.810


0:14:51.810,0:14:53.220
Let's do one more of these.

0:14:53.220,0:14:56.060
The more practice, the[br]better, I think.

0:14:56.060,0:15:01.630
All right, let's say I had[br]negative x squared

0:15:01.630,0:15:06.690
plus 18x, minus 72.

0:15:06.690,0:15:09.320
So once again, I like to factor[br]out the negative 1.

0:15:09.320,0:15:13.040
So this is equal to negative[br]1 times x squared,

0:15:13.040,0:15:16.910
minus 18x, plus 72.

0:15:16.910,0:15:19.650
Now we just have to think of[br]two numbers, that when I

0:15:19.650,0:15:22.330
multiply them I get[br]positive 72.

0:15:22.330,0:15:24.480
So they have to be[br]the same sign.

0:15:24.480,0:15:26.930
And that makes it easier in our[br]head, at least in my head.

0:15:26.930,0:15:29.260
When I multiply them,[br]I get positive 72.

0:15:29.260,0:15:32.040
When I add them, I[br]get negative 18.

0:15:32.040,0:15:34.410
So they're the same sign, and[br]their sum is a negative

0:15:34.410,0:15:36.470
number, they both must[br]be negative.

0:15:36.470,0:15:41.470


0:15:41.470,0:15:43.790
And we could go through all[br]of the factors of 72.

0:15:43.790,0:15:48.510
But the one that springs up,[br]maybe you think of 8 times 9,

0:15:48.510,0:15:53.270
but 8 times 9, or negative 8[br]minus 9, or negative 8 plus

0:15:53.270,0:15:55.210
negative 9, doesn't work.

0:15:55.210,0:15:58.060
That turns into 17.

0:15:58.060,0:15:58.860
That was close.

0:15:58.860,0:15:59.560
Let me show you that.

0:15:59.560,0:16:04.400
Negative 9 plus negative 8, that[br]is equal to negative 17.

0:16:04.400,0:16:06.140
Close, but no cigar.

0:16:06.140,0:16:06.990
So what other ones are there?

0:16:06.990,0:16:08.005
We have 6 and 12.

0:16:08.005,0:16:09.705
That actually seems[br]pretty good.

0:16:09.705,0:16:13.710
If we have negative 6 plus[br]negative 12, that is equal to

0:16:13.710,0:16:15.360
negative 18.

0:16:15.360,0:16:16.590
Notice, it's a bit of an art.

0:16:16.590,0:16:18.810
You have to try the different[br]factors here.

0:16:18.810,0:16:22.360
So this will become negative[br]1-- don't want to forget

0:16:22.360,0:16:29.440
that-- times x minus 6,[br]times x minus 12.

0:16:29.440,0:16:29.866