WEBVTT



00:00:05.000 --> 00:00:12.020
Let us factor this expression. To factor means to write it as a product.

00:00:12.520 --> 00:00:16.640
What do we call this polynomial?

00:00:18.920 --> 00:00:20.120
It is a quadratic binomial.

00:00:25.280 --> 00:00:28.960
Is it missing the quadratic, linear, or constant term?

00:00:33.080 --> 00:00:35.600
It is missing the linear term.

00:00:36.820 --> 00:00:38.940
We could write it like this.

00:00:44.420 --> 00:00:53.340
We need a factor pair whose product is negative 49 and whose sum is zero.

00:00:55.420 --> 00:00:56.500
The pair is negative seven and positive seven.

00:00:56.500 --> 00:00:57.400
plus seven.

00:01:02.460 --> 00:01:04.060
That's plus seven.

00:01:04.740 --> 00:01:07.720
That gives x minus seven and x plus seven.

00:01:09.560 --> 00:01:11.860
Yes. Easy peasy. Wow.

00:01:13.800 --> 00:01:14.700
Okay.

00:01:16.000 --> 00:01:19.700
Do we need to write this step? Definitely not.

00:01:19.700 --> 00:01:22.160
We need two numbers whose product is negative five and whose sum is zero.

00:01:22.160 --> 00:01:29.480
to be negative 5 and add to be 0.

00:01:34.460 --> 00:01:36.480
Maybe it's not.

00:01:37.380 --> 00:01:40.600
Maybe it's not.

00:01:41.280 --> 00:01:43.840
Definitely not.

00:01:44.400 --> 00:01:45.840
What did we...

00:01:56.720 --> 00:01:58.360
Somebody said... What did somebody say?

00:01:59.500 --> 00:02:00.400
What did you say?

00:02:02.380 --> 00:02:02.900
Uh...

00:02:02.900 --> 00:02:03.140
Rad...

00:02:03.140 --> 00:02:04.460
Use square root of five and negative square root of five.

00:02:04.680 --> 00:02:05.660
I'm gonna be a bad man.

00:02:06.760 --> 00:02:07.280
Uh...

00:02:07.280 --> 00:02:08.620
Rad pod and negative red pod.

00:02:13.840 --> 00:02:19.380
Those numbers multiply to negative five and add to zero.

00:02:21.440 --> 00:02:24.080
These factors are not integers, but they are valid real numbers.

00:02:25.300 --> 00:02:30.288
Integers, but they are two numbers that multiplied to be negative five and add to be

00:02:30.288 --> 00:02:31.038
zero

00:02:32.560 --> 00:02:37.640
The factorization is x minus square root of five times x plus square root of five.

00:02:39.820 --> 00:02:42.840
Should we check and see if it works?

00:02:44.140 --> 00:03:00.320
Multiply to check: the middle terms cancel, and the product is x squared minus five.

00:03:01.100 --> 00:03:02.980
Oops, I normally do it here.

00:03:02.980 --> 00:03:06.980
here. And we get minus 5

00:03:10.000 --> 00:03:11.060
at 0

00:03:11.060 --> 00:03:12.180
and that's it.

00:03:16.080 --> 00:03:16.600
Okay.

00:03:16.600 --> 00:03:24.840
Can two real numbers multiply to 36 and add to zero?

00:03:31.700 --> 00:03:33.120
It's not 6 and negative 6.

00:03:33.260 --> 00:03:34.160
Why not?

00:03:36.000 --> 00:03:38.820
Yeah, that does not multiply to be 36.

00:03:42.780 --> 00:03:43.680
Are there any?

00:03:46.120 --> 00:03:47.020
There are none.

00:03:48.660 --> 00:03:53.479
They would need the same sign to have a positive product, so their sum cannot be

00:03:53.479 --> 00:03:54.229
zero.

00:03:55.700 --> 00:03:59.356
Two positives add together, never will be zero. Two negatives add together will

00:03:59.356 --> 00:04:00.200
never be zero.

00:04:01.040 --> 00:04:02.740
Therefore this expression does not factor over the real numbers.

00:04:08.900 --> 00:04:12.140
This is called a sum of squares.

00:04:15.860 --> 00:04:18.260
A sum of squares does not factor over the real numbers.

00:04:18.260 --> 00:04:18.460
Okay.

00:04:18.460 --> 00:04:21.260
The earlier examples are differences of squares.

00:04:21.440 --> 00:04:22.340
What do we call this?

00:04:22.820 --> 00:04:23.720
The difference of squares.

00:04:23.960 --> 00:04:24.860
Difference.

00:04:28.160 --> 00:04:30.600
Oh, squares.

00:04:34.100 --> 00:04:34.680
Okay.

00:04:34.680 --> 00:04:39.600
A difference of two squares always factors.

00:04:44.020 --> 00:04:47.620
A squared minus b squared factors as a minus b times a plus b.

00:04:47.620 --> 00:04:51.120
always factors to what?

00:04:54.240 --> 00:04:56.600
A plus B times

00:04:56.600 --> 00:04:59.700
A minus B.

00:04:59.700 --> 00:05:04.820
and that order doesn't matter.

00:05:06.900 --> 00:05:08.580
Let's call the

00:05:08.580 --> 00:05:10.040
difference of squares.

00:05:33.900 --> 00:05:43.400
Write x squared minus one hundred as x squared minus ten squared.

00:05:43.400 --> 00:05:48.180
It factors as x minus ten times x plus ten.

00:05:48.420 --> 00:05:48.440
OK.

00:05:48.440 --> 00:05:52.840
Rather than multiplying everything out, recognize the square terms.

00:05:52.840 --> 00:05:54.840
and then look for two factors and...

00:05:56.920 --> 00:05:57.820
Too complicated.

00:05:58.680 --> 00:05:59.580
It's very easy.

00:06:00.700 --> 00:06:01.800
Four x squared is the square of two x.

00:06:03.520 --> 00:06:04.420
2x.

00:06:05.120 --> 00:06:06.580
Thirty-six is the square of six.

00:06:07.240 --> 00:06:08.140
Six.

00:06:09.840 --> 00:06:11.360
This is a difference of squares.

00:06:11.760 --> 00:06:12.680
It factors as two x minus six times two x plus six.

00:06:13.080 --> 00:06:14.720
2x minus six.

00:06:15.500 --> 00:06:17.080
That's 2x plus six.

00:06:17.720 --> 00:06:17.880
Done.

00:06:17.880 --> 00:06:23.580
Let's try another one.

00:06:37.660 --> 00:06:40.120
This expression is a difference of squares.

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.

00:06:50.540 --> 00:06:54.000
81 is 9.

00:06:56.080 --> 00:07:00.340
It factors as ten x minus nine times ten x plus nine.

00:07:03.860 --> 00:07:06.660
plus 9

00:07:13.300 --> 00:07:14.700
Uh...

00:07:25.980 --> 00:07:29.440
The square roots are four x and five y.

00:07:30.100 --> 00:07:36.600
So the factors are four x minus five y and four x plus five y.

00:07:37.080 --> 00:07:37.200
Done.

00:07:37.200 --> 00:07:44.300
Okay, let's try another one.

00:08:00.060 --> 00:08:01.200
This is also a difference of squares.

00:08:01.440 --> 00:08:02.720
The square root of this?

00:08:08.920 --> 00:08:11.780
The square root of twenty-five x squared is five x.

00:08:12.000 --> 00:08:14.280
What's the square root of this?

00:08:14.280 --> 00:08:25.140
The square root of seven stays square root of seven, giving conjugate factors.

00:08:25.140 --> 00:08:27.620
This is a fourth-degree binomial rather than a quadratic binomial.

00:08:27.740 --> 00:08:29.760
What would this be?

00:08:29.760 --> 00:08:31.900
4th degree bound.

00:08:38.260 --> 00:08:41.780
However, it is still a difference of squares.

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

00:08:50.392 --> 00:08:50.960
four.

00:08:50.960 --> 00:08:52.740
x squared.

00:08:53.440 --> 00:08:54.440
What squared is 16?

00:08:55.440 --> 00:08:55.820
4.

00:08:55.820 --> 00:08:59.860
4. This is going to be

00:08:59.860 --> 00:09:03.760
First factor as x squared minus four times x squared plus four.

00:09:03.760 --> 00:09:05.580
squared plus 4.

00:09:08.640 --> 00:09:10.760
We are not finished because x squared minus four is another difference of squares.

00:09:13.340 --> 00:09:14.360
Factor x squared minus four again.

00:09:14.360 --> 00:09:16.780
difference of squares.

00:09:19.720 --> 00:09:29.376
The complete factorization is x minus two times x plus two times x squared plus

00:09:29.376 --> 00:09:30.126
four.

00:09:32.340 --> 00:09:39.280
The sum of squares x squared plus four is prime over the real numbers.

00:09:41.380 --> 00:09:48.400
Difference of squares lets us factor higher-degree polynomials.

00:09:54.160 --> 00:09:57.740
And it's going to be very useful when we learn something else later.

00:09:57.740 --> 00:09:59.400
But let's try one more.

00:10:06.840 --> 00:10:08.700
This is another difference of squares.

00:10:15.000 --> 00:10:17.420
What square is x need?

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

00:10:22.440 --> 00:10:23.190
one.

00:10:32.220 --> 00:10:38.440
And we have two fourth degree polynomials. Is one of the two difference the squares?

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

00:11:04.563 --> 00:11:06.100
one.

00:11:07.820 --> 00:11:09.620
We are not done because x squared minus one is another difference of squares.

00:11:10.800 --> 00:11:12.380
No. We got another

00:11:14.100 --> 00:11:15.040
difference of squares.

00:11:29.140 --> 00:11:31.740
Now the expression is completely factored.

00:11:35.140 --> 00:11:36.040
What's that?

00:11:37.680 --> 00:11:39.460
The result is the product of four binomials, and their order does not matter.

00:11:39.460 --> 00:11:42.340
4 binomials. The order does not matter.

00:11:43.580 --> 00:11:47.100
Because the multiplication is commuted. Sorry, let's do one more.

00:11:47.100 --> 00:11:49.040
First factor out the greatest common factor.

00:11:51.660 --> 00:11:52.560
Factor out what?

00:11:53.660 --> 00:11:57.600
Always remove the greatest common factor before applying another factoring method.

00:11:58.800 --> 00:12:00.180
What is the greatest common factor?

00:12:05.700 --> 00:12:07.820
The greatest common factor is ten x cubed.

00:12:15.980 --> 00:12:21.240
The remaining quadratic binomial is a difference of squares.

00:12:42.660 --> 00:12:44.460
Okay, we good?