WEBVTT 00:21.760 --> 00:27.862 In this video, we'll discuss multiplying polynomials. When multiplying a monomial 00:27.920 --> 00:33.634 times a polynomial that is not a monomial, we use the distributive property. Let's look 00:33.680 --> 00:42.676 at these two examples. We're going to begin by multiplying this monomial times this trinomial 00:42.720 --> 00:53.387 using the distributive property. We multiply negative six x times each of these three terms. This will give us negative six x 00:53.421 --> 01:14.909 times x squared plus a negative six x times negative three x plus a negative six x times negative six. So we've 01:14.960 --> 01:23.083 distributed the negative six x to each of these three terms. Now let's simplify the expression. 01:23.120 --> 01:31.959 Negative six x times x squared. x times x squared using the product rule for exponents gives us x to 01:32.000 --> 01:42.269 the one plus two or x to the third power. So this term is negative six x to the third power. Here 01:42.320 --> 01:49.710 we have negative six times negative three which is a positive eighteen, and x times x 01:49.760 --> 01:58.519 is x to the one plus one, or two, x to the second power. So this will be plus eighteen 01:58.560 --> 02:06.894 x to the second power or x squared. And in this third term, we have negative six times negative 02:06.960 --> 02:15.302 six which is a positive thirty-six, and we have x being multiplied times thirty-six, so thirty-six 02:15.360 --> 02:26.680 x. This leaves us with the trinomial negative six x cubed plus eighteen x squared plus thirty-six x. 02:26.720 --> 02:33.787 In this example, we have a monomial a b multiplied times this binomial. Again, 02:33.840 --> 02:40.828 we'll use the distributive property. We're taking a b, multiplying it times the first term which 02:40.880 --> 02:53.407 is two a squared b, and then plus a b multiplied times the second term three a b squared. 02:53.440 --> 03:00.914 Now let's simplify these expressions. a b times two a squared b. 03:00.960 --> 03:08.022 a times a squared is going to be a to the one plus two, or a to the third power. And b times 03:08.080 --> 03:16.997 b is going to be b to the one plus one, or b to the second power. This gives us two a to the third 03:17.040 --> 03:26.273 b to the second. And here in our second term, we have a times a giving us a to the second power, 03:26.320 --> 03:41.722 b times b squared giving us b to the third power. So this term will be three a squared b to the third power. 03:41.760 --> 03:48.996 The distributive property can also be used to multiply binomials. Let's look at this example. 03:49.040 --> 03:56.203 x plus one times x plus three. We have a binomial times a binomial. We'll begin 03:56.240 --> 04:06.580 by distributing this binomial to each of the two terms in the second binomial. That will give us x 04:06.640 --> 04:21.462 times x plus one, and then distributing to the second term gives us plus three times x plus one. 04:21.520 --> 04:31.271 Now simplifying this expression, we'll distribute again. We have a monomial times a binomial. This gives us x times x 04:31.305 --> 04:45.419 which is x squared plus x times one which is x, and distributing three to x plus one gives us three x 04:45.452 --> 04:57.231 plus three. Finally, combining like terms, we see that we have x to the first power here and an x to the first power here. 04:57.280 --> 05:05.639 So we can combine these two terms, one x plus three x to give us four x, and the other two terms 05:05.680 --> 05:18.118 remain the same. So we have x squared plus four x plus three as the product of these two binomials. 05:18.160 --> 05:29.163 Let's continue with a couple of more examples. We're going to multiply x minus six times x plus eight. So we'll distribute 05:29.200 --> 05:41.141 this first binomial to each term in the second binomial. This gives us x times x minus six 05:41.200 --> 05:50.984 plus eight times x minus six. And now again using the distributive property, 05:51.040 --> 06:03.397 this gives us x squared from x times x and minus six x comes from x times a negative six. 06:03.440 --> 06:11.638 Distributing here, we have eight times x, or eight x, and eight times negative six, giving us negative 06:11.680 --> 06:22.282 forty-eight. And finally looking for our like terms, they are here. Negative six x plus eight x 06:22.320 --> 06:29.957 gives us two x. And the other two terms remain the same. They don't have other like terms. So 06:30.000 --> 06:44.271 x squared plus two x minus forty-eight is the product of these two binomials. 06:44.320 --> 06:53.313 In our next example, we have three w minus two times w minus five. Again using the distributive 06:53.360 --> 07:03.557 property, we will distribute three w minus two to each term in the second binomial, giving us w 07:03.600 --> 07:19.639 times three w minus two minus five times three w minus two. And now distributing the monomial 07:19.680 --> 07:33.520 into the binomial gives us w times three w, or three w squared. w times a negative two gives us minus two w. 07:33.553 --> 07:47.934 A negative five times three w gives us minus fifteen w. And a negative five times a negative two gives us plus ten. 07:48.000 --> 07:57.377 Combining like terms here, negative two w minus fifteen w gives us negative seventeen w. 07:57.440 --> 08:09.456 So the final answer here is three w squared minus seventeen w plus ten. 08:09.520 --> 08:17.164 Our next example is four a plus five b times four a minus five b. 08:17.200 --> 08:22.836 Again, let's use the distributive property. We're going to distribute this binomial 08:22.880 --> 08:33.713 to each term in the second binomial, giving us four a times four a plus five b 08:33.760 --> 08:45.625 minus five b times four a plus five b. Using the distributive property again, we have four a 08:45.680 --> 08:56.203 times four a which is sixteen a squared. And four a times five b gives us twenty 08:56.240 --> 09:05.879 a b. Distributing negative five b into this binomial gives us negative five b times four a, 09:05.920 --> 09:15.388 or minus twenty a b. Notice this could be written as b a or a b, but I chose to write it as a 09:15.440 --> 09:23.063 b to show that it is similar to this term here in variables. And we have minus five b 09:23.120 --> 09:34.908 times positive five b giving us minus twenty-five b squared. Now let's combine our like terms. 09:34.960 --> 09:42.282 Positive twenty a b minus twenty a b. When I add positive twenty to negative 09:42.320 --> 09:49.222 twenty I get zero a b, and zero times anything is just zero. So these terms 09:49.280 --> 10:02.602 do not appear in our final answer, leaving us with sixteen a squared minus twenty-five b squared. 10:02.640 --> 10:10.277 In this next example, we have x squared plus one times x squared plus nine. 10:10.320 --> 10:15.215 Distributing x squared plus one to each term in our second binomial 10:15.280 --> 10:30.664 gives us x squared times x squared plus one plus nine times x squared plus one. 10:30.720 --> 10:42.842 Distributing again gives us x squared times x squared which is x to the fourth because two plus two is four. 10:42.880 --> 10:53.787 x squared times one gives us x squared. Nine times x squared gives us nine x squared. 10:53.840 --> 11:03.229 And nine times one gives us nine. And let's look for our like terms. We have an x squared 11:03.280 --> 11:10.737 and a nine x squared. Adding those coefficients together gives us ten x squared. And so this 11:10.800 --> 11:21.781 leaves us with x to the fourth plus ten x squared plus nine as our simplified answer. 11:21.840 --> 11:34.894 Now let's look at our last problem. Write the expression two a plus b squared as a polynomial in standard form. 11:34.960 --> 11:41.067 So we need to think about what this means. When we have an exponential expression, we have a 11:41.120 --> 11:49.242 factor raised to a power, meaning that factor is repeated in multiplication. So we're taking two a 11:49.280 --> 11:57.951 plus b, and we are multiplying it times itself two times. So we want to begin by writing 11:58.000 --> 12:12.365 two a plus b squared as two a plus b times two a plus b. Again, notice that this is a repeated 12:12.400 --> 12:20.440 factor, two a plus b, and we're multiplying it times itself two times because the power is two. 12:20.480 --> 12:27.480 And now we'll use the distributive property. Distributing two a plus b to each term in 12:27.520 --> 12:41.561 the second binomial gives us two a times two a plus b plus b times two a plus b. 12:41.600 --> 12:57.143 And then distributing here, two a times two a is four a squared. Two a times b is plus two a b. 12:57.200 --> 13:04.918 b times two a is also two a b. We can rewrite those terms in whichever order 13:04.960 --> 13:15.161 we like because multiplication is commutative. So we write that as two a b because it's easier to identify this as a like term. 13:15.200 --> 13:25.071 And we have b times b which is b squared. And now looking at our like terms, two a b plus two a b 13:25.120 --> 13:43.323 is four a b. And so this gives us four a squared plus four a b plus b squared as our final answer. 13:43.360 --> 13:49.295 We can use the distributive property to multiply any two polynomials. In this case, 13:49.360 --> 13:56.035 we're going to multiply a binomial times a trinomial. So we'll use the distributive property 13:56.080 --> 14:05.945 and distribute the binomial two x minus five to each of the three terms in this trinomial. 14:06.000 --> 14:15.221 This will give us two x squared times the binomial two x minus five 14:15.280 --> 14:33.640 plus three x times the binomial two x minus five minus five times the binomial two x minus five. 14:33.680 --> 14:40.113 Next, we'll use the distributive property to clear our grouping symbols. 14:40.160 --> 14:48.021 Multiplying two x squared times this binomial gives us two x squared times two x which 14:48.080 --> 14:59.699 is four x to the third power. And two x squared times minus five gives us minus ten x squared. 14:59.760 --> 15:07.140 Distributing three x to each of these two terms in this binomial gives us three x times two x 15:07.200 --> 15:18.684 which is plus six x squared. And three x times a negative five gives us minus fifteen x. 15:18.720 --> 15:26.926 Distributing the minus five to the binomial to each term gives us negative five times two x, 15:26.960 --> 15:37.537 or minus ten x. And negative five times negative five gives us plus twenty-five. 15:37.600 --> 15:45.945 Finally, we're going to combine like terms. My first term is four x to the third power. 15:46.000 --> 15:54.587 There are no other x to the third powers, so there are no like terms to combine to four x cubed. My 15:54.640 --> 16:03.062 second term is negative ten x squared, and it can be combined with positive six x squared. 16:03.120 --> 16:11.637 Negative ten x squared plus six x squared is minus four x squared. 16:11.680 --> 16:19.479 Negative fifteen x can be combined with negative ten x because they are like terms. 16:19.520 --> 16:32.825 Negative fifteen x minus ten x is minus twenty-five x. And plus twenty-five does not have any like terms to be added to it. 16:32.880 --> 16:42.201 This is the final product of this multiplication of these two polynomials. 16:42.240 --> 16:46.906 In this example, we're going to write the expression t plus three to the third 16:46.960 --> 16:55.381 power as a polynomial in standard form. Let's recall what this expression means. The factor t 16:55.440 --> 17:02.922 plus three is being multiplied times itself three times because the exponent is three. 17:02.960 --> 17:14.600 So that means that t plus three to the third power can be rewritten as t plus three times t plus three 17:14.640 --> 17:21.541 times t plus three. So we have a binomial being multiplied times itself 17:21.600 --> 17:28.681 three times. We're going to begin this problem by multiplying these first two binomials to 17:28.720 --> 17:35.621 each other, simplifying that product, and then multiplying that resulting polynomial 17:35.680 --> 17:42.828 times the third binomial. So let's go ahead and multiply t plus three times t plus three 17:42.880 --> 17:54.974 using the distributive property. We get t times t plus three plus three 17:55.040 --> 18:03.783 times t plus three. And this product of these two binomials will then be multiplied 18:03.840 --> 18:13.059 times the third binomial t plus three. Let's simplify this expression by using the distributive 18:13.120 --> 18:25.805 property to clear the grouping symbols. t times t plus three gives us t squared plus three t. 18:25.840 --> 18:37.350 And then three times t plus three gives us three t plus nine. And this polynomial will be multiplied times the 18:37.383 --> 18:48.294 binomial t plus three after we simplify. So we can still combine like terms here in this product to be able 18:48.327 --> 19:02.675 to simplify this polynomial prior to multiplying by t plus three. Three t plus three t is six t. So this gives us t squared 19:02.720 --> 19:11.884 plus six t plus nine. This is the product of these first two binomials, and then we're multiplying 19:11.920 --> 19:20.826 this times the third binomial t plus three. Let's use the distributive property again. Distributing 19:20.880 --> 19:33.406 this trinomial to each of the two terms in the binomial. This will give us t times the trinomial 19:33.440 --> 19:44.350 which is t squared plus six t plus nine and then plus three times the trinomial 19:44.400 --> 19:56.428 again which is t squared plus six t plus nine. Now we'll distribute again to clear these grouping symbols. 19:56.480 --> 20:16.015 Distributing t to each of these three terms in this trinomial gives us t to the third power plus six t squared plus nine t. 20:16.080 --> 20:31.163 And distributing this three to each term in this trinomial gives us three t squared plus eighteen t 20:31.200 --> 20:41.240 plus twenty-seven. And finally we need to look for like terms that can be combined. 20:41.280 --> 20:51.083 The first term is a t to the third power, and there are no other t cubed expressions. So there are no like terms to combine 20:51.120 --> 20:58.524 with the first term. I have a six t squared as my next term, and I notice that I have 20:58.560 --> 21:12.204 a plus three t squared here. Six t squared plus three t squared gives me nine t squared. 21:12.240 --> 21:21.780 My next term is a plus nine t. And I have an eighteen t here. So these like terms 21:21.840 --> 21:35.794 can be combined to give me twenty-seven t. And my last term is a plus twenty-seven. 21:35.840 --> 21:42.201 There are no other constants to be added to this. So there are no like terms to be added at this 21:42.240 --> 21:57.920 point. And our final answer is t cubed plus nine t squared plus twenty-seven t plus twenty-seven.