WEBVTT 00:21.280 --> 00:27.628 In this video, we will learn about adding and subtracting polynomials. A polynomial is a type of 00:27.680 --> 00:35.703 algebraic expression. Specifically, a polynomial in x is a finite sum of terms in the form a x to 00:35.760 --> 00:45.079 the n where a is a real number and n is a whole number. This is an example of a polynomial. This 00:45.120 --> 00:55.055 polynomial has four terms. Each of these terms can be written in the form a x to the n. A polynomial 00:55.120 --> 01:08.436 whose powers are written in descending order as this one is is known to be in standard form. 01:08.480 --> 01:15.376 The numerical coefficient in each of these terms is known as the coefficient. 01:15.440 --> 01:26.487 For the first term x to the third, the coefficient is implied. There is a one as the numerical factor in front of x cubed. 01:26.520 --> 01:49.476 So the coefficient is one. For this term negative two x squared, the coefficient is negative two. For the term three x, 01:49.520 --> 01:59.887 the coefficient is three. And for the term five, notice that five is a coefficient, but 01:59.920 --> 02:14.335 because it doesn't have a variable in this term, we call this the constant term or just constant. 02:14.400 --> 02:23.711 An example of an expression that is not a polynomial would be x to the negative two 02:23.760 --> 02:31.785 plus eight x minus nine, and the reason this is not a polynomial is because the exponent 02:31.840 --> 02:43.397 is not a whole number. Each term of a polynomial has a degree. The degree of a term is the sum of 02:43.440 --> 02:51.071 the exponents on the variables contained in the term. The degree of a polynomial is the 02:51.120 --> 02:58.846 greatest degree of any term of the polynomial. Let's take a look at a couple of examples. 02:58.880 --> 03:07.321 Determine if the expression is a polynomial, and if it is a polynomial, state the degree. 03:07.360 --> 03:15.696 Our first expression is x squared minus eight x to the fifth plus six x to the third plus nine x 03:15.760 --> 03:23.470 minus five. To determine whether this expression is a polynomial, we look at the exponents on the 03:23.520 --> 03:33.314 variables. We notice that these are whole numbers, and so this is a polynomial. 03:33.360 --> 03:39.620 Next, to find the degree we look for the degree of each term, and we identify what the highest 03:39.680 --> 03:48.829 degree of the terms are. This degree is two. The degree of this term is five. 03:48.880 --> 03:58.205 This is three. The degree here is one. The degree here is zero, and that's because we can write this 03:58.240 --> 04:06.046 as negative five x to the zero power. The largest degree of all of these terms 04:06.080 --> 04:19.560 is five. So five is the degree of the polynomial. So yes, this is a polynomial of degree five. 04:19.600 --> 04:27.668 Notice that this polynomial is not written in standard form because the powers are not in descending order. 04:27.701 --> 04:38.445 The next example we have is x plus x to the one-half minus six. Let's look at the powers on the variables. We 04:38.479 --> 05:02.436 have one, one-half, and zero. Since one-half is not a whole number, this is not a polynomial. 05:02.480 --> 05:06.907 Remember polynomials can contain more than one variable as long as the 05:06.960 --> 05:13.380 exponents on the variables are whole numbers. In these examples, we'll determine the degree 05:13.440 --> 05:19.319 of the polynomial expression, and then we're going to state that the polynomial is a monomial, 05:19.360 --> 05:29.062 binomial, or trinomial or none of these. What does that mean? A monomial is a polynomial with 05:29.120 --> 05:37.137 one term. Mono meaning one. Binomial is a polynomial with two terms, 05:37.200 --> 05:44.344 and a trinomial is a polynomial with three terms. If there are more terms than three, 05:44.400 --> 05:50.284 we don't have a special classification for that. We just call it a polynomial. 05:50.320 --> 05:58.926 This first example is negative two x squared y plus seven x y cubed. 05:58.960 --> 06:10.437 There are one, two terms in this polynomial meaning that this is a binomial. 06:10.480 --> 06:13.941 Now let's look at the degree. To determine the degree, 06:14.000 --> 06:21.081 we look at the degree of each term which is the sum of the powers on the variables. 06:21.120 --> 06:30.757 The power on the x is two, and the power on the y is one. So this term has a degree of three. Here 06:30.800 --> 06:39.633 the power on the x is one, and the power on the y is three. So the degree of this term is four. That 06:39.680 --> 06:52.579 means that the degree of the polynomial is four because that is the largest degree of each term. 06:52.640 --> 07:03.156 Now let's look at this polynomial. We have three a squared plus two a b plus b squared. 07:03.200 --> 07:15.636 There are one, two, three terms meaning this is a trinomial. Now let's look at the degree of each term. 07:15.669 --> 07:26.346 The power on the variable here is two. The power on the a is one, and the power on the b is one, meaning that the degree 07:26.400 --> 07:34.421 of the second term is one plus one or two. And the power on the b in this third term 07:34.480 --> 07:52.439 is two. So all three terms have a degree of two, meaning that the degree of the polynomial is two. 07:52.480 --> 07:59.312 Polynomials that contain like terms can be simplified by combining those like terms. 07:59.360 --> 08:07.954 Recall that like terms mean that the terms have the same variables raised to the same powers. 08:08.000 --> 08:15.228 And when you combine like terms, you add or subtract the coefficients to those terms. 08:15.280 --> 08:22.502 Our first example is three b squared minus eight b squared plus twelve b. 08:22.560 --> 08:30.444 Let's look for the like terms. This term has a b squared and this term has a b squared, 08:30.480 --> 08:38.118 the same variables raised to the same power. So they're like terms. This term is not a like term 08:38.160 --> 08:46.193 to the other two because it has a b raised to the first power. So when I combine these like terms, 08:46.240 --> 08:55.302 we add or subtract the coefficient. So three minus eight, giving us negative five b 08:55.360 --> 09:05.479 squared. The variable and its power remain the same. And we still have the plus twelve b. This 09:05.520 --> 09:15.021 polynomial is simplified. It is also in standard form because the powers are in descending order. 09:15.055 --> 09:23.063 Let's take a look at our next example. One-half x y plus three x squared y squared plus 09:23.120 --> 09:32.439 three-halves x y minus x squared y squared. Let's identify the like terms. Here we have 09:32.480 --> 09:39.946 x and y both raised to the first power, and here we also have a term with x and y raised to the 09:40.000 --> 09:48.655 first power. So these will be like terms, exact same variables raised to the exact same powers. 09:48.720 --> 09:55.395 We also have the second term which has an x squared y squared, and the fourth term has 09:55.440 --> 10:03.303 an x squared y squared. Same variables raised to the same powers means that those two terms 10:03.360 --> 10:11.945 are like terms. Using the commutative property, I can rearrange these terms 10:12.000 --> 10:20.520 to make it simpler when I add them or subtract them. So I have one-half x y plus three-halves 10:20.560 --> 10:34.501 x y plus three x squared y squared minus x squared y squared. Now focusing on our first two 10:34.560 --> 10:41.541 terms here, which are like terms, we will add the coefficients. One-half plus three-halves 10:41.600 --> 10:51.084 is four-halves, and four-halves is the same as two. So this gives us two x y 10:51.120 --> 10:57.457 combining these first two terms. And then combining the last two terms, 10:57.520 --> 11:05.398 we have a coefficient of three and a coefficient of minus one. So combining three plus a negative 11:05.440 --> 11:14.674 one gives us two, and the variables stay the same. So we have plus two x squared y squared. 11:14.720 --> 11:25.218 These two terms cannot be combined because they are not like terms. 11:25.280 --> 11:32.926 In these next examples, we will add the polynomials. To add polynomials, we simply combine 11:32.960 --> 11:40.667 like terms. We have the polynomial three x squared plus eight x minus five that we are adding to 11:40.720 --> 11:48.341 the polynomial two x squared minus six. Let's rearrange this polynomial using the Commutative 11:48.400 --> 11:54.414 Property of Addition and the Associative Property of Addition, and we'll do some regrouping putting 11:54.480 --> 12:04.024 our like terms next to each other for clarity. So three x squared and two x squared are like terms, 12:04.080 --> 12:14.434 so I can rewrite this as three x squared plus two x squared. And then I have a positive eight x. So 12:14.480 --> 12:21.074 plus eight x. And there is no other like term to eight x, so I will just put eight x. 12:21.120 --> 12:32.519 And I have a minus five minus six. So I will add negative five and negative six. 12:32.560 --> 12:39.225 Now adding these like terms here, we keep the variable and the power, and we add the coefficient, 12:39.280 --> 12:51.838 giving us five x squared. Here we have plus eight x. And negative five minus six is negative eleven. So this 12:51.871 --> 13:04.751 gives us minus eleven. This is our new polynomial that is the sum of the original two polynomials. 13:04.800 --> 13:14.260 In our next example, we are adding this trinomial to this trinomial. And again, let's rearrange this 13:14.320 --> 13:28.675 so that we have the like terms next to each other for clarity. Six a and two a are like terms. 13:28.720 --> 13:40.119 We have a minus seven b and a plus nine b, so we'll add a negative seven b and nine b together. 13:40.160 --> 13:55.134 And we have a positive three and a positive five, and so we'll add three and five. Six a plus two a gives us eight a. 13:55.200 --> 14:10.016 Negative seven b plus nine b gives us plus two b. And three plus five gives us eight. 14:10.080 --> 14:17.040 To subtract a polynomial, we are going to add its opposite. So let's look at these examples 14:17.056 --> 14:25.565 where we are subtracting polynomials. Here we have a binomial minus another binomial. So we'll 14:25.600 --> 14:36.442 rewrite this first as adding the opposite. So we have nine x squared plus four y squared, and 14:36.480 --> 14:43.483 instead of subtracting this polynomial, we're going to add the opposite of this polynomial 14:43.520 --> 14:57.030 which is this polynomial times negative one. I'm going to write it this way first. 14:57.063 --> 15:06.472 So we're adding the opposite of the given polynomial. Well, notice when I have negative one times this polynomial, 15:06.506 --> 15:15.348 what this does is change the sign of all of the terms in these grouping symbols. And so this becomes nine 15:15.381 --> 15:28.361 x squared plus four y squared. Minus one times minus two x squared gives me plus two x squared, and minus one 15:28.394 --> 15:40.106 times ten y squared gives us minus ten y squared. And so now you can see this as adding two polynomials or adding two 15:40.160 --> 15:47.040 binomials. And combining like terms now, we see that we have nine x squared plus 15:47.046 --> 16:00.720 two x squared. That will give us eleven x squared. And four y squared minus ten y squared. Four 16:00.727 --> 16:10.737 minus ten is minus six. So this will give us minus six y squared. Notice that when we add or subtract 16:10.800 --> 16:18.745 polynomials, we end up with a polynomial. In this case, we ended up with a binomial. 16:18.800 --> 16:26.285 As we look at this next subtraction problem, we have a polynomial with four terms 16:26.320 --> 16:35.027 minus a polynomial with three terms. Again, to subtract we just simply add the opposite. 16:35.061 --> 16:46.672 So we can write this as nine t cubed plus t squared minus twelve t minus four. And now I'm 16:46.720 --> 16:52.912 going to add the opposite of this polynomial. I'm going to do it a little more efficiently than I 16:52.960 --> 16:59.719 did here. Remember adding the opposite means that this polynomial is multiplied by negative one 16:59.760 --> 17:08.261 which changes all of the signs of these terms. So this will become plus a negative t cubed, 17:08.320 --> 17:19.806 or minus t cubed. Plus a negative of negative t squared, so this will become plus t squared. 17:19.840 --> 17:27.246 This term will become minus fifteen t. Again take a closer look, all of these 17:27.280 --> 17:34.687 terms have opposite signs in this addition problem when I rewrote the subtraction 17:34.720 --> 17:43.162 to addition. And now let's combine like terms. We have a nine t cubed, 17:43.200 --> 17:52.240 and I have a negative t cubed. Those can be combined to give me eight t cubed. I have a 17:52.271 --> 18:04.750 t squared and another positive t squared. One t squared plus one t squared gives me two t squared. 18:04.800 --> 18:14.827 I have a minus twelve t and a minus fifteen t. Minus twelve plus minus fifteen 18:14.880 --> 18:21.868 gives me minus twenty-seven. So this term will be minus twenty-seven t. 18:21.920 --> 18:29.475 And I have a minus four still remaining with no other constant terms available. 18:29.520 --> 18:39.785 And so this is the polynomial answer, or result, of subtracting these two given polynomials. 18:39.840 --> 18:47.159 In this last example, we're going to simplify this expression where we add and or subtract binomials. 18:47.193 --> 18:57.270 Five c minus four d is being added to this binomial four c plus eight d, and then we are subtracting this binomial 18:57.303 --> 19:06.579 two d minus c. Recall that subtracting is adding the opposite. So let's rewrite this problem 19:06.640 --> 19:18.758 as five c minus four d plus four c plus eight d. And now we're going to rewrite this as an addition of the opposite, 19:18.791 --> 19:39.245 so plus a negative two d plus c. And this gives us five c minus four d plus four c plus eight d minus two d 19:39.280 --> 19:48.921 plus c. That's a long expression. Let's find our like terms. I have five c, four 19:48.960 --> 20:02.668 c, and plus c. Five plus four plus one gives us ten, so this term will be ten c. 20:02.720 --> 20:14.347 And now we have minus four d plus eight d minus two d. Minus four plus eight 20:14.400 --> 20:33.840 minus two gives me two d, plus two d. So our final answer is the binomial ten c plus two d.