WEBVTT Kind: captions Language: en 00:00:00.656 --> 00:00:04.476 >> Welcome to the Cypress College Math Review on Multiplying Polynomials. 00:00:05.906 --> 00:00:08.136 Objective 1: Types of Polynomials. 00:00:08.546 --> 00:00:11.876 A monomial is a polynomial with one term. 00:00:12.206 --> 00:00:20.246 Some examples of a monomial would be 7x squared, negative 3ab, and negative x cubed y squared. 00:00:20.936 --> 00:00:23.956 A binomial is a polynomial with two terms. 00:00:24.526 --> 00:00:31.116 Some examples of a binomial would be 6x minus 7, and 3a squared plus 2b. 00:00:31.716 --> 00:00:35.506 And finally a trinomial is a polynomial with three terms. 00:00:36.046 --> 00:00:43.226 And an example of a trinomial would be negative 3x squared y plus 5xy minus 2. 00:00:46.256 --> 00:00:49.056 Objective 2: Multiply Two Monomials. 00:00:49.926 --> 00:00:55.246 To multiply two monomials, the first step will be to multiply their coefficients together. 00:00:55.616 --> 00:00:59.206 Then the second step will be to use the product rule of exponents 00:00:59.306 --> 00:01:02.196 to find the exponent of the corresponding variables. 00:01:02.536 --> 00:01:08.126 So recall that the Product Rule of Exponents tells us that for multiplying two things 00:01:08.126 --> 00:01:12.646 that have the same base, we are able to add their exponents together. 00:01:13.486 --> 00:01:17.356 So let's go ahead and multiply these two monomials together. 00:01:17.766 --> 00:01:20.476 3x squared times 5x cubed. 00:01:21.036 --> 00:01:25.446 So the first thing that we should do is multiply their coefficients together, 00:01:25.576 --> 00:01:34.946 so then we would multiply together 3 times 5 and then we'll be multiplying the variables together 00:01:35.346 --> 00:01:40.556 so then we will be multiplying together x squared times x cubed. 00:01:42.166 --> 00:01:45.586 OK, so multiplying 3 times 5 that will give us 15. 00:01:46.826 --> 00:01:51.536 And then now looking at the variables notice that we're multiplying two things 00:01:51.536 --> 00:01:55.966 that have the same base of x, so then according to the product rule 00:01:55.966 --> 00:01:59.866 of exponents we're able to add their exponents together. 00:01:59.866 --> 00:02:07.376 So then we get a final answer of 15x to the 5th. 00:02:12.276 --> 00:02:15.906 For our next example we'll multiply the two monomials -- 00:02:16.176 --> 00:02:20.776 negative 2x squared y by 7x cubed y squared. 00:02:21.396 --> 00:02:26.186 So again, the first thing that we'll do is multiply the coefficients together. 00:02:26.586 --> 00:02:32.566 Negative 2 times 7 and then notice here we have both x's and y's. 00:02:32.916 --> 00:02:41.496 So first we'll multiply the x's together and then we'll also multiply the y's together. 00:02:45.296 --> 00:02:50.396 So from here multiplying negative 2 times 7, that will give us negative 14. 00:02:51.046 --> 00:02:54.666 And then again using the product rule of exponents we'll be able 00:02:54.666 --> 00:02:57.916 to add the exponents together for the x's. 00:02:58.726 --> 00:03:04.006 And then looking at the y's, this first y looks like it doesn't have an exponent 00:03:04.126 --> 00:03:07.396 but there's actually an invisible exponent of 1 there. 00:03:07.866 --> 00:03:12.846 So again similarly we'll be able to add the exponents of the y's together as well. 00:03:15.200 --> 00:03:23.020 So then from here we get a final answer of negative 14x to the 5th, y to the 3rd. 00:03:25.260 --> 00:03:27.900 Pause the video and try these problems. 00:03:32.320 --> 00:03:36.180 Objective 3: Multiply Any Polynomial by a Monomial. 00:03:36.716 --> 00:03:41.616 To multiply a polynomial by a monomial, you will use the distributive property 00:03:41.846 --> 00:03:45.866 to multiply the monomial by each term of the polynomial. 00:03:46.086 --> 00:03:52.146 So for this example we want to multiply the monomial 3x to the trinomial 2x 00:03:52.146 --> 00:03:55.876 to the 4th minus 6x cubed plus 3x squared. 00:03:56.246 --> 00:04:03.566 So to multiply these two together, we will distribute the 3x to every term in the trinomial 00:04:04.166 --> 00:04:07.356 and when we distribute what we will be doing is multiplying 00:04:07.356 --> 00:04:11.356 that monomial 3x to every term in the trinomial. 00:04:12.046 --> 00:04:14.146 So let's go ahead and distribute. 00:04:15.356 --> 00:04:20.146 So distributing we will get 3x times 2x to the 4th 00:04:20.546 --> 00:04:27.506 and then we have a minus 3x times 6x to the third. 00:04:27.906 --> 00:04:33.446 And then finally plus 3x times 3x squared. 00:04:35.006 --> 00:04:38.306 Next step is to just multiply these monomials together. 00:04:38.456 --> 00:04:42.386 So looking at the first two monomials, multiplying those two. 00:04:42.986 --> 00:04:50.906 3 times 2 gives us 6x times x to the 4th, again adding the exponents will give us x to the 5th. 00:04:51.976 --> 00:04:53.856 Looking at the next two monomials -- 00:04:54.256 --> 00:05:00.696 negative 3 times 6 that will give us a minus 18, x times x to the 3rd. 00:05:00.906 --> 00:05:04.816 Again, adding the exponents will give us x to the 4th. 00:05:05.476 --> 00:05:11.826 And then finally looking at the last two monomials, 3 times 3 will give us plus 9. 00:05:12.376 --> 00:05:17.766 And x times x squared adding the exponents will give us x cubed. 00:05:24.066 --> 00:05:28.426 For this example we want to multiply the monomial negative 2a squared b times the 00:05:28.426 --> 00:05:33.366 binomial 4a cubed b squared minus 3a b squared. 00:05:33.886 --> 00:05:39.706 So to multiply we will distribute the monomial to both terms in the binomial. 00:05:39.856 --> 00:05:43.896 Again, when we distribute what we're really doing is multiplying the monomial 00:05:43.896 --> 00:05:46.086 to every term in the binomial. 00:05:46.696 --> 00:05:55.356 So distributing will get negative 2a squared b times 4a cubed b squared. 00:05:55.966 --> 00:06:05.746 Then we have a minus, distributing again we'll have negative 2a squared b times 3ab squared. 00:06:06.216 --> 00:06:08.866 And all there's left to do now is to multiply. 00:06:09.546 --> 00:06:14.996 So multiplying the negative 2a squared b times the 4a cubed b squared, 00:06:15.506 --> 00:06:19.636 negative 2 times 4 gives us negative 8. 00:06:20.046 --> 00:06:25.996 a squared times a cubed, adding the exponents of the a's will give us a to the 5th. 00:06:26.786 --> 00:06:32.106 And then b times b squared adding the exponents of the b's will give us b cubed. 00:06:32.886 --> 00:06:37.496 Notice here that we have two negatives, so then this will be come a plus. 00:06:37.996 --> 00:06:45.496 So then 2 times 3 will give us 6, a squared times a adding the exponents 00:06:45.496 --> 00:06:49.616 of the a's will give us a cubed and then b times b squared, 00:06:49.760 --> 00:06:52.980 adding the exponents of the b's will give us b cubed. 00:06:55.280 --> 00:06:57.920 Pause the video and try these problems. 00:07:03.080 --> 00:07:05.620 Objective 4: Multiply Two Binomials. 00:07:06.016 --> 00:07:11.706 To multiply two binomials we will use the distributive property twice or a common acronym 00:07:11.706 --> 00:07:15.726 that we use to multiply two binomials together is the acronym FOIL. 00:07:16.186 --> 00:07:20.126 Where F will tell us to multiply the First terms together. 00:07:21.056 --> 00:07:24.776 O will tell us to multiply the Outer terms together. 00:07:25.756 --> 00:07:29.026 I will tell us to multiply the Inner terms together. 00:07:29.366 --> 00:07:33.336 And then L tells us to multiply the Last terms together. 00:07:36.100 --> 00:07:42.736 So here if you want to multiply the two binomials x plus 7 times 2x minus 3, 00:07:42.796 --> 00:07:44.286 we will follow FOIL. 00:07:44.696 --> 00:07:47.866 So F tells us to multiply the first terms together, 00:07:47.866 --> 00:07:54.206 so then we'll multiply together x times 2x to give us 2x squared. 00:07:55.076 --> 00:08:00.586 Then we should multiply the outer terms together that would be x times negative 3 00:08:00.876 --> 00:08:09.906 to give us minus 3x and then we want to multiply the inner terms together 7 times 2x will give us 00:08:09.906 --> 00:08:15.806 a positive 14x and then we want to multiply together the last terms, 00:08:16.076 --> 00:08:21.456 positive 7 times a negative 3 will give us negative 21. 00:08:22.636 --> 00:08:26.906 And then from here notice that we have some like terms that we can combine -- 00:08:27.116 --> 00:08:30.886 the two middle terms -- the negative 3x and the 14x, 00:08:30.956 --> 00:08:34.356 we can combine by adding their coefficients together. 00:08:35.056 --> 00:08:39.926 So then from here we'll get 2x squared then combining the negative 3x 00:08:40.066 --> 00:08:46.666 with the 14x will give us plus 11x and then minus 21. 00:08:49.676 --> 00:08:56.926 For this example we'll multiply together the two binomials 3x minus 5y times 2x plus 7y. 00:08:57.366 --> 00:09:01.156 So since we're multiplying two binomials together, we will follow FOIL. 00:09:01.156 --> 00:09:04.816 So F will tell us to multiply the first terms together. 00:09:05.176 --> 00:09:11.716 So then we'll multiply together 3x times 2x to give us 6x squared. 00:09:12.556 --> 00:09:17.226 And then we should multiply the outer terms together 3x times 7y. 00:09:17.466 --> 00:09:21.166 That will give us plus 21xy. 00:09:22.076 --> 00:09:24.786 And then we should multiply the inner terms together. 00:09:24.946 --> 00:09:31.496 The negative 5y times a positive 2x will give us negative 10xy 00:09:32.216 --> 00:09:38.136 and then we should multiply the last terms together the negative 5y times the positive 7y 00:09:38.416 --> 00:09:42.276 to give us negative 35y squared. 00:09:42.966 --> 00:09:46.496 And then notice we have two like terms that we can combine. 00:09:46.756 --> 00:09:51.106 The 21xy and the negative 10xy are like terms. 00:09:51.416 --> 00:09:54.976 So we can combine those two by adding their coefficients together. 00:09:55.626 --> 00:10:03.026 So from here we'll get 6x squared combining the two like terms we'll get plus 11xy 00:10:03.600 --> 00:10:07.120 and then we have minus 35y squared. 00:10:08.400 --> 00:10:11.040 Pause the video and try these problems. 00:10:16.040 --> 00:10:19.620 Objective 5: Multiply the Sum and Difference of the Same Two Terms. 00:10:20.006 --> 00:10:26.526 For here, let's say we want to multiply together a plus b times a minus b. Notice 00:10:26.526 --> 00:10:28.756 that these two binomials are very similar, 00:10:28.866 --> 00:10:32.196 except one has a plus and the other one has a minus. 00:10:32.426 --> 00:10:34.026 So let's see what will happen here. 00:10:34.726 --> 00:10:38.306 So since we're multiplying two binomials we will follow FOIL. 00:10:38.696 --> 00:10:42.936 So we should multiply the first terms together to get a squared, 00:10:43.836 --> 00:10:47.956 multiplying the outer terms together we'll get minus ab, 00:10:49.026 --> 00:10:53.236 multiplying the inner terms together we'll get plus ab 00:10:53.236 --> 00:10:58.386 and then multiplying the last terms together we'll get minus b squared. 00:10:59.226 --> 00:11:03.896 So looking at what we have, notice again those two middle terms are like terms 00:11:04.176 --> 00:11:06.246 and we can combine those two together. 00:11:06.716 --> 00:11:10.666 But when we combine those two together they're just going to eliminate each other, 00:11:11.136 --> 00:11:15.636 so then we're just left with a squared minus b squared. 00:11:16.396 --> 00:11:20.286 So when you're multiplying two binomials of this form -- 00:11:20.366 --> 00:11:27.516 a plus b, times a minus b that's just going to multiply to become a squared minus b squared. 00:11:29.006 --> 00:11:35.986 For this example we want to multiply together x plus 4 times x minus 4. 00:11:36.506 --> 00:11:42.856 So notice that this is of the form a plus b times a minus b. So then we'll know 00:11:42.976 --> 00:11:46.866 that this will multiply to become a squared minus b squared. 00:11:47.716 --> 00:11:50.216 So then we just need to identify what is our a? 00:11:50.796 --> 00:11:52.066 And what is our b? 00:11:52.646 --> 00:12:00.466 So notice here that our a will be equal to x, and our b will be equal to 4. 00:12:02.536 --> 00:12:09.106 So x squared just becomes x squared minus 4 squared gives us 16. 00:12:12.096 --> 00:12:18.686 Here we want to multiply 2x plus 5y times 2x minus 5y. 00:12:19.106 --> 00:12:24.306 So notice that this is of the form a plus b times a minus b. So we know 00:12:24.306 --> 00:12:28.606 that this will multiply to be of the form a squared minus b squared. 00:12:28.836 --> 00:12:34.376 So again we just have to identify what our a is and what our b is. 00:12:35.146 --> 00:12:42.136 So for this example our a will become 2x and our b is 5y. 00:12:45.036 --> 00:12:53.646 So 2x, all of that squared becomes 4x squared minus and then squaring 5y, 00:12:54.020 --> 00:12:57.980 5 squared is 25, and then we have y squared. 00:12:59.500 --> 00:13:02.140 Pause the video and try these problems. 00:13:06.240 --> 00:13:09.040 Objective 6: Squaring a Binomial. 00:13:09.466 --> 00:13:13.786 So let's say we have the binomial a plus b, which we wish to square. 00:13:14.056 --> 00:13:16.846 So if you want to square the binomial a plus b, 00:13:17.126 --> 00:13:22.026 that will just be multiplying a plus b times a plus b. 00:13:22.026 --> 00:13:27.336 So since we're multiplying two binomials we will follow FOIL. 00:13:27.536 --> 00:13:32.016 So we should multiply the first terms together to get a squared, 00:13:32.996 --> 00:13:38.526 then we should multiply the outer terms together to get plus ab, 00:13:38.736 --> 00:13:43.256 then we should multiply the inner terms together to get plus ab, 00:13:43.256 --> 00:13:48.436 and then multiplying the last terms together will get plus b squared. 00:13:49.496 --> 00:13:54.836 Now notice those two terms in the middle are like terms so we are able to combine those. 00:13:55.186 --> 00:14:02.186 So then we get a squared plus 2ab plus b squared. 00:14:02.936 --> 00:14:07.106 So if you have something of the form a plus b quantity squared, 00:14:07.406 --> 00:14:13.516 we know that that will multiply out to be a squared plus 2ab plus b squared. 00:14:15.796 --> 00:14:22.016 Now let's do a similar example but now we want to square a minus b. So if you want 00:14:22.016 --> 00:14:27.736 to square a minus b that is the same as just multiplying a minus b, 00:14:28.286 --> 00:14:32.526 times a minus b, and again we will follow FOIL. 00:14:32.896 --> 00:14:36.636 So multiplying the first terms together we get a squared, 00:14:37.216 --> 00:14:41.156 multiplying the outer terms together we get minus ab, 00:14:42.286 --> 00:14:46.006 multiplying the inner terms together we get minus ab 00:14:46.006 --> 00:14:51.876 and then multiplying the last terms together we get plus b squared. 00:14:52.346 --> 00:14:55.026 And again we have two terms that are like terms, 00:14:55.326 --> 00:15:03.736 so combining those two middle terms we get a squared minus 2ab plus b squared. 00:15:04.386 --> 00:15:09.786 So if you have something of the form a minus b quantity squared, that will multiply 00:15:09.786 --> 00:15:14.116 out to be a squared minus 2ab plus b squared. 00:15:17.756 --> 00:15:21.876 For this example we wish to square the binomial a plus 7. 00:15:22.276 --> 00:15:26.026 So notice this is of the form a plus b quantity squared. 00:15:26.436 --> 00:15:32.616 So we know that this will multiply out to be a squared, plus 2ab, plus b squared. 00:15:33.646 --> 00:15:38.076 So then we just need to figure out what our a is and what our b is. 00:15:38.856 --> 00:15:47.316 For this example our a is just going to be a. And our b is 7. 00:15:51.326 --> 00:15:59.196 So simplifying here we'll get a squared plus 2 times 7 gives us 14a 00:15:59.556 --> 00:16:03.716 and then plus 7 squared gives us 49. 00:16:08.716 --> 00:16:12.966 For this example we wish to square the binomial 3x minus 4. 00:16:13.136 --> 00:16:18.906 So notice that this is of the form a minus b quantity squared, so this will multiply 00:16:18.946 --> 00:16:23.656 out to be a squared minus to 2ab plus b squared. 00:16:24.276 --> 00:16:28.846 So to figure out this problem, again all we have to do is identify what is our a 00:16:28.846 --> 00:16:40.136 and what is our b. So for this one our a is 3x and our b is 4. 00:16:42.866 --> 00:16:48.306 So simplifying, 3x squared, well 3 squared is 9. 00:16:48.306 --> 00:16:51.056 And then x squared is just x squared. 00:16:51.506 --> 00:16:58.946 And then we have minus 2 times 3 gives us 6 times 4 gives us 24. 00:16:58.946 --> 00:17:05.920 So then we have minus 24x plus 4 squared will give us plus 16. 00:17:08.040 --> 00:17:10.680 Pause the video and try these problems. 00:17:15.040 --> 00:17:18.260 Objective 7: Multiply Any Two Polynomials. 00:17:18.446 --> 00:17:23.976 To multiply any two polynomials, the first step will be to multiply each term 00:17:23.976 --> 00:17:28.096 of the first polynomial to every term in the second polynomial. 00:17:28.596 --> 00:17:31.646 Then step two will be to combine any like terms. 00:17:31.836 --> 00:17:38.436 So if you wish to multiply together 2x plus 5 times x squared plus 4x minus 3, 00:17:38.436 --> 00:17:44.396 we will first have to distribute the 2x to every term in the second polynomial. 00:17:44.556 --> 00:17:49.406 And then we will distribute the 5 to every term in the second polynomial. 00:17:49.986 --> 00:17:53.206 So again, the first thing we should do is distribute the 2x 00:17:53.626 --> 00:17:56.476 to every term in the second polynomial. 00:17:57.046 --> 00:18:09.946 So doing that 2x times x squared will give us 2x cubed, 2x times 4x will give us plus 8x squared. 00:18:10.646 --> 00:18:15.946 And then 2x times negative 3 will give us negative 6x. 00:18:16.376 --> 00:18:22.236 So now that we're done distributing the 2x, the next step is to distribute the 5 00:18:22.576 --> 00:18:25.196 to every term in the second polynomial. 00:18:25.636 --> 00:18:31.526 So then 5 times x squared will give us plus 5x squared. 00:18:32.016 --> 00:18:36.776 5 times 4x will give us plus 20x. 00:18:37.446 --> 00:18:42.336 And then 5 times a negative 3 will give us minus 15. 00:18:43.336 --> 00:18:47.976 So now that we're done distributing, now we're going to combine any like terms. 00:18:48.936 --> 00:18:51.436 So there are no like terms with the 2x cubed. 00:18:51.556 --> 00:18:52.836 So I'll just bring him down. 00:18:54.816 --> 00:18:59.076 But here with the 8x squared that's like terms with the 5x squared. 00:18:59.416 --> 00:19:04.326 So I can combine those two to get plus 13x squared 00:19:05.516 --> 00:19:09.626 and then the negative 6x is like terms with the 20x. 00:19:09.756 --> 00:19:14.336 So then I can combine those two to get plus 14x. 00:19:14.340 --> 00:19:17.720 And then lastly we have that minus 15. 00:19:21.200 --> 00:19:23.840 Pause the video and try these problems. 00:19:29.040 --> 00:19:31.680 Objective 8: Multiply More than Two Polynomials. 00:19:32.236 --> 00:19:38.676 To multiply more than two polynomials all we'll do is just multiply two polynomials at a time. 00:19:39.316 --> 00:19:43.916 So notice for this example we want to multiply together three polynomials. 00:19:44.256 --> 00:19:48.446 But again all we're going to do is just work with two polynomials at a time. 00:19:48.926 --> 00:19:51.226 So the first two that we're going to be working 00:19:51.226 --> 00:19:56.096 with will be multiplying together x plus 1, times x plus 2. 00:19:56.716 --> 00:20:00.976 So notice that's just multiplying together two binomials, so we're going to FOIL. 00:20:01.756 --> 00:20:06.656 So first thing we do is multiply the first terms together to get x squared. 00:20:07.406 --> 00:20:12.086 And then we multiply the outer terms together to get plus 2x. 00:20:12.706 --> 00:20:16.446 Multiplying the inner terms together we get plus x 00:20:16.666 --> 00:20:21.016 and then multiplying the last terms together we get plus 2. 00:20:21.416 --> 00:20:27.736 Hey but then remember we still have to multiply that to the 2x minus 5. 00:20:28.686 --> 00:20:34.276 So before we multiply, looking at that first polynomial we can simplify it a little bit more 00:20:34.276 --> 00:20:39.076 since we have some like terms there we can combine the 2x with the x, 00:20:39.426 --> 00:20:46.206 so doing that first we'll get x squared plus 3x plus 2. 00:20:46.206 --> 00:20:50.926 And then that we'll multiply to the 2x minus 5. 00:20:51.666 --> 00:20:55.126 So now if we wish to multiply these two polynomials together, 00:20:55.126 --> 00:21:00.086 what we'll have to do is distribute each term of the first polynomial 00:21:00.326 --> 00:21:02.886 to every term in the second polynomial. 00:21:03.416 --> 00:21:07.476 So the first thing that we'll do is multiply the x squared 00:21:07.756 --> 00:21:10.896 to the each term, in that second polynomial. 00:21:11.976 --> 00:21:16.826 So x squared times 2x gives us 2x cubed, 00:21:17.396 --> 00:21:23.276 then x squared times negative 5 will give us minus 5x squared. 00:21:23.756 --> 00:21:30.436 So then the next step will be to distribute the 3x to every term in the second polynomial. 00:21:31.026 --> 00:21:37.146 So then 3x times 2x will give us plus 6x squared 00:21:37.146 --> 00:21:43.216 and then 3x times a negative 5 will give us minus 15x. 00:21:44.036 --> 00:21:46.806 And then the last thing we have to distribute is the 2. 00:21:46.806 --> 00:21:56.376 So then we'll distribute the 2 to every term in the second polynomial, so 2 times 2x gives us 4x 00:21:56.946 --> 00:22:01.076 and then 2 times negative 5 will give us negative 10. 00:22:02.706 --> 00:22:07.206 So now that we're done distributing, all we have to do is combine like terms. 00:22:07.966 --> 00:22:13.376 So there are no like terms to combine with the 2x cubed, so we'll just bring that down. 00:22:14.216 --> 00:22:18.956 But here the negative 5x squared is like terms with the 6x squared. 00:22:19.256 --> 00:22:23.756 So combining those two we'll get plus x squared 00:22:24.286 --> 00:22:28.696 and then the negative 15x is like terms with the 4x. 00:22:29.080 --> 00:22:35.860 So then that will combine to give us negative 11x, and then we have the minus 10. 00:22:38.920 --> 00:22:41.560 Pause the video and try these problems.