WEBVTT 00:21.520 --> 00:27.395 In this video, we'll be simplifying algebraic expressions. To begin with, we need to identify 00:27.440 --> 00:34.735 parts of an algebraic expression. A term is a number, a variable, or the product of a number 00:34.800 --> 00:42.476 and one or more variables raised to powers. These are examples of terms: negative four, 00:42.510 --> 00:52.686 x, negative four x, negative four x to the third power, six x cubed y squared, six x 00:52.720 --> 01:02.830 cubed over y squared, three-fifths a squared b c to the fourth. Each of these is a term. 01:02.880 --> 01:13.140 The coefficient of a term is the numerical factor of the term. Negative four is also a coefficient. 01:13.200 --> 01:19.713 The coefficient of this term is one. There is an implied one in front of the x. The 01:19.760 --> 01:29.056 coefficient of this term three-fifths a squared b c to the fourth is three-fifths. 01:29.120 --> 01:36.597 Terms with the same variable raised to the same powers are called like terms. If a term 01:36.640 --> 01:42.837 is not a like term, it's called an unlike term. Let's take a look at this example. 01:42.880 --> 01:50.444 Consider the term two-fifths a squared b c to the fourth power. 01:50.480 --> 02:05.326 What is the coefficient? In this term, the coefficient is two-fifths. 02:05.360 --> 02:12.266 Now think about what a like term might look like. A like term would have the same variables raised 02:12.320 --> 02:19.807 to the same power, but it could have a different numerical coefficient. So instead of two-fifths, 02:19.840 --> 02:32.119 perhaps it's three. Three a squared b c to the fourth would be an example of a like term. 02:32.160 --> 02:40.027 Any numerical coefficient in front of a squared b c to the fourth would work to create a like term. 02:40.080 --> 02:46.066 An unlike term could have any numerical coefficient, but it has to have either a 02:46.100 --> 02:56.277 different variable or a variable raised to a different power. So an example of an unlike term 02:56.320 --> 03:05.386 would be three a b c. Notice I have the same variables, but instead of an a to the second 03:05.440 --> 03:14.895 power, I have an a to the first power. And right away that makes it an unlike term. 03:14.960 --> 03:20.034 An algebraic expression containing the sum or difference of like terms 03:20.080 --> 03:27.541 can be simplified by applying the Distributive Property. This is called combining like terms. 03:27.600 --> 03:35.215 For example, three x plus two x. This can be rewritten using the Distributive Property as 03:35.280 --> 03:48.429 three plus two times x. Three plus two is five. This would give us five x. Let's do some examples. 03:48.480 --> 03:55.402 Simplify the expression by combining like terms. In our first example, we have negative y squared 03:55.440 --> 04:01.642 plus five y squared. Using the Distributive Property, we could rewrite this as negative 04:01.680 --> 04:19.626 one plus five times y squared. This would give us four times y squared, or four y squared. 04:19.680 --> 04:29.002 In the next example, we have fourteen a minus eight a plus seven. Let's identify where the like terms are. Fourteen 04:29.040 --> 04:36.443 a and negative eight a are like terms because they have the same variable raised to the same power. 04:36.480 --> 04:50.357 So using the Distributive Property, we can rewrite this as fourteen minus eight times a plus seven. 04:50.400 --> 05:02.503 This gives us fourteen minus eight which is six a plus seven. Since six a and seven are unlike terms, they cannot be 05:02.536 --> 05:13.747 combined. This is our simplified answer. In this next example, we have five a plus one plus four 05:13.780 --> 05:23.857 a minus nine. Let's identify our like terms. We have five a and four a which are like terms. Using the Commutative Property, 05:23.920 --> 05:29.563 I can rearrange this expression so that the five a and four a are next to each other. 05:29.600 --> 05:35.802 One and negative nine are also like terms. So let's use the Commutative Property. This 05:35.840 --> 05:45.312 is equal to five a plus four a plus one minus nine. And now using the Distributive Property, 05:45.360 --> 05:57.391 we have five plus four times a, and we can combine one minus nine to get negative eight. 05:57.440 --> 06:08.368 Five plus four is nine. So this gives us nine a minus eight. Nine a and minus eight are unlike terms, 06:08.402 --> 06:15.943 so this is our simplified answer. In the last expression, we have ten x plus three x squared 06:16.000 --> 06:23.884 plus two x minus x squared. Again, we begin by identifying like terms, the same variable 06:23.920 --> 06:30.591 raised to the same power. So even though there is an x here in this term and an x in this term, 06:30.640 --> 06:38.832 they are not like terms because they do not have the same power. Ten x and two x are like terms. 06:38.880 --> 06:46.840 I'll use the Commutative Property to put them next to each other, for clarity. And three x squared 06:46.880 --> 06:56.750 and minus x squared are also like terms. They have the same variable raised to the same power. 06:56.800 --> 07:05.058 Now using the Distributive Property, I have ten plus two times x 07:05.120 --> 07:23.076 plus three minus one times x squared, giving us a final answer of twelve x plus two x squared. 07:23.120 --> 07:29.082 When simplifying algebraic expressions, we often use the Distributive Property twice. 07:29.120 --> 07:35.322 Once, to remove the parentheses that are in the given expression and then again to combine like 07:35.360 --> 07:42.362 terms. Let's look at these examples. The directions are to simplify the expression. 07:42.400 --> 07:47.067 First, we'll use the Distributive Property to remove the parentheses. 07:47.120 --> 07:56.343 So negative four gets multiplied times three x and times the positive two, giving us 07:56.400 --> 08:07.320 negative twelve x minus eight. And the rest of the expression is plus eight minus ten x. 08:07.360 --> 08:13.860 And now to simplify this expression, we again will consider the Commutative Property and 08:13.920 --> 08:19.800 the Distributive Property. Now I’m not going to rewrite the expression this time, but we know that 08:19.840 --> 08:26.273 I could use the Commutative Property to rewrite it and put this minus ten x next to the minus twelve x. 08:26.320 --> 08:33.246 I'll just move into using the Distributive Property. So I have minus twelve and minus ten. 08:33.280 --> 08:42.656 So minus twelve minus ten times x, and minus eight plus eight is zero. 08:42.720 --> 08:52.199 Here, I have minus twelve minus ten, which is negative twenty-two, x. 08:52.240 --> 08:58.605 This is our simplified expression. In this next example, I have two sets of 08:58.640 --> 09:05.145 parentheses. So we will use the Distributive Property twice to remove the parentheses. 09:05.200 --> 09:14.821 Three times two p is six p, and three times negative nine is negative twenty-seven. 09:14.880 --> 09:22.095 Here, we have a minus sign in front of these parentheses which is just the same as multiplying 09:22.128 --> 09:32.339 or having a negative one in front of those parentheses. So I am distributing a negative one into the parentheses p plus ten. 09:32.372 --> 09:45.552 This will give us negative p minus ten. Now combining our like terms, we have a six p and a minus p that are like terms. 09:45.600 --> 09:50.824 And so using the Distributive Property, I can write that as six minus one 09:50.880 --> 09:58.765 times p. And I have a minus twenty-seven and a minus ten remaining. That combines 09:58.800 --> 10:07.774 to give us a negative thirty-seven, or minus thirty-seven. I have now six minus one which is 10:07.807 --> 10:23.023 five p. Six minus one which is five times p minus thirty-seven. This is our simplified expression. 10:23.056 --> 10:28.595 In our last example, we have one set of grouping symbols. We have the minus sign in front of the 10:28.640 --> 10:35.802 grouping symbol, so there is an implied one, or minus one, in front of those parentheses. 10:35.840 --> 10:46.579 So I will distribute a negative one into these parentheses, giving us a negative five x y, 10:46.640 --> 10:55.555 negative one times two z is negative two z, and I still have a plus three z 10:55.600 --> 11:02.762 minus nine x y. Let's look for our like terms. I'm looking for the same variables 11:02.800 --> 11:11.705 raised to the same powers. So I have an x y here and an x y here. These are like terms. 11:11.760 --> 11:20.280 Using the Distributive Property with those two terms, I have minus five minus nine times x y. 11:20.320 --> 11:26.119 And then I also have like terms here, z to the first power and z to the first power. 11:26.160 --> 11:36.496 So using the Distributive Property, I have plus a negative two plus three z. 11:36.560 --> 11:46.539 Now I have minus five minus nine, which is negative fourteen, then times x y. 11:46.573 --> 12:01.421 And negative two plus three is one, so this will be one z, or just z. This is our final, simplified expression. 12:01.454 --> 12:09.229 In these next examples, we are going to write the sentence as an algebraic expression, 12:09.280 --> 12:17.470 and then simplify. In our first example, we are going to add four x plus two to six 12:17.520 --> 12:34.587 x plus five. So we're beginning with six x plus five and adding four x plus two. 12:34.640 --> 12:41.294 Now because addition is commutative, the order in which I added these expressions didn't matter. I 12:41.360 --> 12:51.304 could have begun with a four x plus two or written this as four x plus two plus six x plus five. 12:51.360 --> 12:57.076 Now to simplify. Because the Associative Property and the Commutative Property both apply to 12:57.120 --> 13:04.751 addition, I can drop the parentheses. And now I can apply the Distributive Property. Combining my 13:04.800 --> 13:19.065 like terms six x plus four x, this would give me six plus four x, and then I have five plus two, 13:19.098 --> 13:29.375 which we'll add in our next step. Here, I have six plus four, which is ten, times x 13:29.440 --> 13:38.351 plus seven. So our final simplified expression is ten x plus seven. 13:38.400 --> 13:47.393 In our next example, we are to subtract four x plus two from six x plus five. So it's important 13:47.440 --> 13:53.866 to note here that we are beginning with six x plus five, and we are subtracting four x plus 13:53.920 --> 14:09.782 two from it. Now here the order is very important because subtraction is not commutative. It is only 14:09.840 --> 14:19.325 commutative if I were to rewrite a subtraction problem as addition of the opposite sign. 14:19.360 --> 14:24.764 Now I'm going to clear the parentheses by utilizing the Distributive Property. 14:24.800 --> 14:31.704 In this first set of parentheses, I'm multiplying by one. So there is nothing to do other than to 14:31.760 --> 14:38.745 rewrite the expression six x plus five. With this second set of parentheses, there is an implied 14:38.800 --> 14:44.350 one in front of the parentheses. And so we are going to distribute a negative one 14:44.400 --> 14:54.360 into the parentheses, giving us negative four x minus two. And now using the Distributive 14:54.400 --> 15:07.640 Property, I can rewrite this problem as six minus four times x plus five minus two. 15:07.680 --> 15:20.686 This gives me two x plus three, and this is our final, simplified expression. 15:20.720 --> 15:28.594 When presented with a problem, it is often useful to translate phrases into algebraic expressions. 15:28.640 --> 15:34.434 So let's take a look at these first two examples. The directions are to write the phrase as an 15:34.480 --> 15:43.075 algebraic expression. In this first example, we have one-third of a number decreased by two. So 15:43.120 --> 15:53.386 we'll begin with one-third of a number. Let's call that number x. This would be one-third x, and 15:53.440 --> 16:01.861 we are decreasing that one-third of a number by two. Decreasing by two means that we are subtracting 16:01.920 --> 16:12.205 two. One-third x minus two is our algebraic expression that is represented by this phrase. 16:12.240 --> 16:18.177 In the next example, we have nine times the sum of a number and three. 16:18.240 --> 16:23.950 So we're going to start with the sum of a number and three. Again, let's let that number be 16:24.000 --> 16:34.760 represented by x. So the sum of a number and three is x plus three. We need nine times that sum, 16:34.800 --> 16:50.176 so that would look like nine times x plus three in parentheses. Nine times the sum of x plus three. 16:50.240 --> 16:55.615 Let's look at this example. A plot of land is in the shape of a rectangle. 16:55.680 --> 17:03.389 If one side is four x minus two meters and one side is five x minus six meters, express the 17:03.440 --> 17:13.299 perimeter of the lot as a simplified expression in x. Let's draw a sketch to represent this problem. 17:13.360 --> 17:23.309 The shape is a rectangle, and one of the sides is four x minus two. I don't know whether that's 17:23.360 --> 17:27.947 the short side or the long side, but for the purpose of the sketch it really doesn't matter. 17:28.000 --> 17:35.788 So I will put this side as four x minus two. And one side is five x minus six. 17:35.840 --> 17:40.826 Well, I know that in a rectangle opposite sides have the same length, 17:40.880 --> 17:51.804 so this side here has to still be four x minus two. So the five x minus six I will put here. 17:51.840 --> 17:57.643 Now to write the perimeter, I know the perimeter means the length around, 17:57.680 --> 18:06.919 which perimeter of a rectangle can be represented as two times the length plus two times the width. 18:06.960 --> 18:17.296 Or it could be the sum of each of the four sides. So if I want to write an expression for the perimeter, I'm going to 18:17.330 --> 18:33.946 calculate two L plus two W. That would be two times four x minus two plus two times five x minus six. This expression 18:34.000 --> 18:39.885 represents the perimeter. Now let's simplify it. We're going to use the Distributive Property 18:39.920 --> 18:50.496 first to clear the parentheses. This will give us eight x minus four plus ten x minus twelve. 18:50.560 --> 19:01.707 And now combining like terms, again we'll use the Distributive Property, combining eight plus ten x. 19:01.760 --> 19:17.156 And I have minus four minus twelve. And so this would give us eighteen x, and minus four minus twelve, minus sixteen. 19:17.200 --> 19:37.840 So an expression for our perimeter is eighteen x minus sixteen meters.