WEBVTT 00:21.840 --> 00:27.628 In this video, we're going to discuss special products. We're going to begin with 00:27.680 --> 00:40.908 the product of two binomials, a plus b times a plus b, which can be written as a plus b squared. 00:40.960 --> 00:48.182 This product can be visualized geometrically using the area model for a square. 00:48.240 --> 01:00.494 In this square, this side length is a plus b and a plus b. So the sum of each of these four areas 01:00.560 --> 01:14.675 will give us the product of a plus b squared. The area of the square a plus b squared is the same 01:14.720 --> 01:26.987 as a plus b times a plus b. Now let's look at the area of each of these rectangles. 01:27.040 --> 01:35.863 In this top area, we have a side length of a and another side length of a. So this is a square. And 01:35.920 --> 01:47.474 the area of this square would be a times a or a squared. In this next shape, we have a rectangle. 01:47.520 --> 01:54.682 It has a side length of b and another side length of a, and so the area of this rectangle would be 01:54.720 --> 02:04.992 the product of these two sides. We will write that as a b which also could be written as b a. 02:05.040 --> 02:15.870 In this upper rectangle, we have a side length of a and a side length of b. So again we have a b. 02:15.920 --> 02:27.548 And in this bottom portion, we have a square with side length b and b, so this area is b squared. 02:27.600 --> 02:46.200 The area of the entire larger square a plus b times a plus b is the sum of these four areas. This gives us a squared. 02:46.240 --> 02:58.679 We have an a b here and an a b here. Adding those two like terms together will give us two a b. 02:58.720 --> 03:15.396 And we have this square, which is b squared. So the product of a plus b squared is a squared plus two a b plus b squared. 03:15.440 --> 03:25.706 This leads us to two identities, a plus b squared which is written here 03:25.760 --> 03:33.040 and also similarly we could write a minus b squared, and what would change here is the 03:33.047 --> 03:50.264 sign of the two a b. We get a squared minus two a b plus b squared for a minus b squared. 03:50.320 --> 03:57.471 Let's do some examples. In this first example, we have three x plus five squared. 03:57.520 --> 04:04.978 Using our formula, a plus b quantity squared, we know that this product will give us 04:05.040 --> 04:15.789 a squared plus two a b plus b squared. So this is a, and this is b. And a squared would be 04:15.840 --> 04:36.844 three x squared. Then we have two times the product of a and b plus b squared. 04:36.880 --> 04:51.158 And now we just have to simplify this expression. Three x squared will give us nine x squared. 04:51.200 --> 04:56.430 Two times three times five gives us thirty, and we're multiplying 04:56.480 --> 05:07.141 that times x so we have thirty x. And five squared is twenty-five. 05:07.200 --> 05:18.519 So the product three x plus five squared is nine x squared plus thirty x plus twenty five. 05:18.560 --> 05:26.593 In our next example, we have four m minus one squared. Notice the minus sign here. And so what 05:26.640 --> 05:35.869 changes in the formula that is different from a plus b squared is the a b term or the two a b term 05:35.920 --> 05:45.846 which will be negative. So we begin with a squared, which will be four m squared. 05:45.879 --> 06:05.165 Minus two a b, two times four m times one. Plus b squared, plus one squared. 06:05.200 --> 06:20.781 Simplifying this expression, we get sixteen m squared. Here we have negative two times four m times one. 06:20.814 --> 06:32.259 This will be a negative or minus eight m. And one squared is one, so plus one. 06:32.320 --> 06:47.407 Four m minus one squared is equal to sixteen m squared minus eight m plus one. And our last example is two times x minus 06:47.441 --> 07:00.287 two y squared. We need to work the product of x minus two y squared first. So we'll keep our two on the 07:00.320 --> 07:10.764 outside of our grouping symbols, and we'll use our formula to simplify x minus two y squared. 07:10.800 --> 07:19.940 We begin with a squared, which will be x squared. Minus two times a times b, 07:20.000 --> 07:34.655 so that will be minus two times x times two y. And then plus b squared, which will give us plus two y 07:34.720 --> 07:57.077 squared. Let's simplify what are in these brackets. This gives us x squared minus. This gives us four x y, 07:57.120 --> 08:07.554 and then plus two y squared is plus four y squared. And finally, we'll distribute our two 08:07.600 --> 08:23.937 to clear the grouping symbols, giving us two x squared minus eight x y plus eight y squared. 08:24.000 --> 08:31.478 Another special product is for the sum and difference of the same two terms. 08:31.520 --> 08:37.784 In this example, we have two x plus one times two x minus one. Let's 08:37.840 --> 08:43.790 use the distributive property first to multiply this out and see what happens. 08:43.840 --> 08:53.233 If I distribute this first binomial to each of the two terms in the second binomial, we get two x 08:53.280 --> 09:13.587 times two x plus one minus one times two x plus one. Distributing here gives us four x squared plus two x. 09:13.620 --> 09:26.433 And distributing here gives us minus two x minus one. Now combining like terms, you can see that what happens is that 09:26.466 --> 09:39.546 our linear terms here, two x and negative two x, will cancel out, leaving us with four x squared minus one. 09:39.600 --> 09:45.785 Whenever we multiply a sum and difference of the same two terms, 09:45.840 --> 10:02.669 this pattern will continue to happen. This is called the difference of two squares. 10:02.720 --> 10:07.941 The reason why it's called that is that you can see that the product 10:08.000 --> 10:19.553 here leads us with a difference, a subtraction, of two perfect squares. The formula for the difference of two squares is 10:19.586 --> 10:36.903 a plus b times a minus b gives us a squared minus b squared, the difference of two squares. 10:36.960 --> 10:44.744 Let's look at a few examples. First we have three x plus five times three x minus five. 10:44.800 --> 10:51.151 Notice that we have the same two terms three x and five showing up in both 10:51.200 --> 10:58.525 of these binomials with a difference of signs. So we'll use our difference of two squares formula. 10:58.560 --> 11:08.735 This will give us a squared minus b squared, where a is three x, so we'll have three x squared, 11:08.800 --> 11:23.550 and b is five, so we'll have minus five squared. Simplifying here gives us nine x squared. 11:23.600 --> 11:36.696 And minus five squared gives us minus twenty-five, the difference of two squares. In our next example, we have 11:36.730 --> 11:45.305 ten a plus b times ten a minus b. Again, we'll use our difference of two squares formula. 11:45.360 --> 12:00.353 This will give us the first terms squared, or ten a squared, minus the second term squared, 12:00.400 --> 12:12.198 which is minus b squared. Ten a squared gives us one hundred a squared, and we have minus b 12:12.240 --> 12:21.474 squared. So our product is one hundred a squared minus b squared. 12:21.520 --> 12:29.616 And in our third example, we have x squared minus seven times x squared plus seven. 12:29.680 --> 12:37.057 We'll use our difference of two squares formula, and we'll multiply or square the first term, 12:37.120 --> 12:51.538 x squared squared, minus the second term squared. Remember our exponent rules when I 12:51.600 --> 13:18.000 have a power raised to a power I multiply. So this will give us x to the fourth minus forty-nine.