WEBVTT 00:21.680 --> 00:28.320 In this section, we're going to factor binomials. Specifically, we're going to look at an expression 00:28.329 --> 00:34.902 called the difference of two squares. A binomial is a difference of two squares 00:34.960 --> 00:42.176 if the first term can be written as a squared, or a perfect square, and the second term 00:42.240 --> 00:48.115 can be written as b squared, and there is a subtraction between these two terms. 00:48.160 --> 00:56.657 a squared minus b squared is a difference of two squares, and it can be factored into a plus b 00:56.720 --> 01:02.663 times a minus b. We learned in a previous section that if I use the distributive property 01:02.720 --> 01:12.606 to multiply a plus b times a minus b that I would end up with this difference of two squares. 01:12.640 --> 01:19.146 x squared minus nine is an example of a binomial that is a difference of two squares because the 01:19.200 --> 01:26.120 first term is a perfect square, the second term is a perfect square, and there is a difference or 01:26.160 --> 01:35.396 a subtraction between these two terms. Thus, it factors into x plus three times x minus three. 01:35.440 --> 01:41.869 Let's take a look at some examples. We want to factor the binomial completely for each of 01:41.920 --> 01:50.511 these expressions. b squared minus twenty-five is our first example. We notice that the first term 01:50.560 --> 01:57.618 is a perfect square, b squared, and twenty-five is a perfect square. It can be written as five 01:57.680 --> 02:13.067 squared. And we are subtracting them. Thus, this can be written as b plus five times b minus five. 02:13.120 --> 02:24.278 b squared minus five squared gives me b plus five times b minus five. 02:24.320 --> 02:31.318 In our second example, we have eighty-one x squared minus sixteen. We notice first that our 02:31.360 --> 02:39.793 first term can be written as a perfect square. Eighty-one x squared can be written as nine x 02:39.840 --> 02:48.102 squared. And then sixteen can be written as four squared, and there is a subtraction between them. 02:48.160 --> 02:56.744 So now we recognize this as the difference of two squares that can be written in factored form as 02:56.800 --> 03:14.762 a, being nine x, plus b, which is four, times nine x minus four. a plus b times a minus b. 03:14.800 --> 03:26.040 In our next example, we have four x squared minus forty-nine y squared. The first term can be written or is a perfect square, 03:26.073 --> 03:39.086 and it can be written as two x squared. This last term is a perfect square and can be written as seven y 03:39.120 --> 03:45.059 squared. And there's a subtraction between these two terms. So this is the difference 03:45.120 --> 03:58.105 of two squares. Thus, it factors into two x plus seven y times two x minus seven y. And 03:58.160 --> 04:06.747 remember the order of these factors does not matter because multiplication is commutative. 04:06.800 --> 04:14.021 Let's look at our last example. Fifty x squared minus two. Doesn't look like this 04:14.080 --> 04:20.661 is a difference of two squares in this form. But remember we must first look for a common factor, 04:20.720 --> 04:28.202 if possible, and I notice that fifty x squared and two have a common factor of two. So we'll 04:28.240 --> 04:38.112 begin by factoring out the GCF of two which leaves us with twenty-five x squared minus one. 04:38.160 --> 04:45.040 And this binomial is the difference of two squares because twenty-five x squared is the 04:45.052 --> 04:56.363 same as five x squared and one is the same as one squared. And so this binomial factors into five 04:56.400 --> 05:09.443 x plus one times five x minus one. And don't forget we still have the common factor two that belongs in this 05:09.476 --> 05:23.323 factored form as well. And so this binomial factors into two times five x plus one times five x minus one. 05:23.360 --> 05:30.497 Let's continue by factoring these binomials completely. Our next example is sixty-four n 05:30.560 --> 05:37.704 squared minus thirty-six. You may recognize this as a difference of two squares, 05:37.760 --> 05:43.644 but first remember that we always want to begin our factoring by factoring out a common factor, 05:43.680 --> 05:51.618 or a GCF, if possible first. That will make our factoring more efficient, so looking at these two 05:51.680 --> 06:01.562 terms sixty-four n squared and thirty-six, the GCF of these two terms is four. So if I factor 06:01.600 --> 06:13.407 a four out of each term, that will leave me with sixteen n squared minus nine. Sixteen n squared 06:13.440 --> 06:21.315 is a perfect square, and nine is a perfect square, and there is a subtraction between these two terms, 06:21.360 --> 06:28.422 meaning that this is the difference of two squares and can be factored into a plus b 06:28.480 --> 06:37.798 times a minus b. So our final factored form will include the four as our GCF 06:37.840 --> 06:49.076 and then a plus b where a is four n and b is three. So four n plus three, 06:49.120 --> 07:00.020 and a minus b which would be four n minus three. So our completely factored form of this binomial 07:00.080 --> 07:17.804 is four times four n plus three times four n minus three. Our next example is sixty-four n squared plus thirty-six. Let's begin 07:17.840 --> 07:24.178 also by looking for a common factor. You may notice that this example looks very 07:24.240 --> 07:30.517 similar to the one that we just did. The only difference here of these two examples is the 07:30.560 --> 07:41.228 sign. So we know that we can factor a GCF of four from these terms. This gives us four 07:41.280 --> 07:51.138 times sixteen n squared plus nine. Now looking at the binomial that remains here, 07:51.200 --> 07:59.079 we have a perfect square and a perfect square, but this is not the difference of two squares 07:59.120 --> 08:07.187 because there is a sum or a plus between these two terms. The sum of two squares cannot be factored. 08:07.220 --> 08:20.200 So this is our completely factored form of this binomial. Our next example is y to the fourth minus eighty-one. 08:20.233 --> 08:30.544 There is no common factor between these two terms, but I can recognize that each of these terms is a perfect square. 08:30.577 --> 08:44.524 y to the fourth can be written as y squared squared, and eighty-one can be written as nine squared, 08:44.560 --> 08:50.664 and there is a subtraction between these two terms making this a difference of two squares. 08:50.720 --> 09:07.314 So this difference of two squares factors into y squared plus nine times y squared minus nine. Now let's look a little closer 09:07.360 --> 09:17.224 at these two binomials that remain. y squared plus nine is the sum of two squares. We just 09:17.280 --> 09:25.065 talked about the fact that we cannot factor the sum of two squares in the real number system. 09:25.120 --> 09:32.739 y squared minus nine can be factored it is a difference of two squares and can be factored 09:32.800 --> 09:41.715 as y plus three times y minus three. So our final factored form of y to the fourth minus 09:41.760 --> 09:54.094 eighty-one will be y squared plus nine times y plus three times y minus three. 09:54.160 --> 10:04.037 And our last example is negative t cubed plus t. I have two terms. I want to begin by looking 10:04.080 --> 10:11.144 for the GCF of these two terms, and when my expression begins with a negative recall that 10:11.200 --> 10:23.724 it's often convenient to factor out a negative. So we will factor out a common factor of negative t. 10:23.760 --> 10:35.235 This will leave us with a positive t squared and a minus one. You can check that GCF factoring by redistributing 10:35.280 --> 10:41.074 through your grouping symbols. And now I recognize that the binomial that remains, t squared minus 10:41.120 --> 10:51.785 one, is the difference of two squares, and it can be factored into t plus one times t minus one. 10:51.840 --> 11:13.840 And let's not forget our GCF of negative t. So our final factored form is negative t times t plus one times t minus one.