0:00:11.229,0:00:14.383 College Algebra students, [br]Dillon here. 0:00:16.705,0:00:21.495 We're going to do a little quick [br]wrap up before Section 3.3. 0:00:21.495,0:00:23.484 Here we go. 0:00:31.160,0:00:39.498 We're studying polynomials, and we’ve [br]learned there's a connection between 0:00:41.900,0:00:50.351 the x-intercepts (which we also call zeros) [br]and factors of the polynomial. 0:00:52.330,0:00:54.339 An important theorem— 0:00:54.339,0:00:57.033 We've learned to do [br]synthetic division also. 0:00:57.033,0:01:01.472 That's what we're going to apply here [br]that we're learning in Section 3.3. 0:01:01.472,0:01:05.337 We have this factor theorem[br]that says “c is a zero“— 0:01:05.337,0:01:12.174 or x-intercept of the polynomial— [br]“P if and only if x minus c is a factor.” 0:01:12.174,0:01:16.767 So, if c is a zero, this [br]polynomial has to be a factor. 0:01:16.767,0:01:20.254 If this is a factor, [br]then it has to be a zero. 0:01:20.254,0:01:23.952 That's what “if and only if” means: [br]it works in both directions. 0:01:23.952,0:01:26.618 Let's see an application of this. 0:01:27.733,0:01:29.946 Let P of x, be this[br]following polynomial. 0:01:29.946,0:01:33.107 We're going to show, first of all[br]that P of 1 equals zero. 0:01:33.107,0:01:37.259 Now, according to this factor, [br]if we could show it does equal zero, 0:01:37.259,0:01:41.512 then x minus 1 would [br]have to be a factor. 0:01:41.512,0:01:44.835 Then we can use that to [br]actually factor the polynomial. 0:01:44.835,0:01:47.259 Let me show you what[br]I'm talking about. 0:01:47.259,0:01:50.974 First of all, let's just check[br]is P of 1 equal to zero. 0:01:50.974,0:01:55.061 We're going to have 1 cubed[br]minus 7 times 1 plus 6. 0:01:55.305,0:02:00.157 We have 1 plus 6 is 7, [br]7 minus 7, is zero, check. 0:02:05.460,0:02:09.521 Okay, so if it's zero,[br]then what has to be true? 0:02:09.521,0:02:13.447 Then x minus 1 [br]must be a factor. 0:02:15.579,0:02:20.442 We can factor it by using our new skill [br]that we learned: synthetic division. 0:02:20.442,0:02:22.224 Remember, the factor— 0:02:22.224,0:02:26.028 if the binomial that we're trying[br]to divide by is x minus 1, 0:02:26.028,0:02:28.340 we put a little 1 [br]here in the corner. 0:02:28.340,0:02:32.753 We fill in the coefficients including [br]a zero if we’re missing any terms 0:02:32.753,0:02:34.746 (and we are missing[br]the squared term). 0:02:34.746,0:02:37.627 The coefficient on [br]x to the third is 1. 0:02:37.627,0:02:41.549 And then we'll have a zero, [br]minus 7, and 6. 0:02:42.290,0:02:44.706 And draw the box[br]below the 6. 0:02:44.706,0:02:46.553 And then we apply [br]synthetic division. 0:02:46.553,0:02:50.113 Bring down the 1, [br]1 times 1 is 1. 0:02:50.113,0:02:52.330 Add here, we get 1. 0:02:52.330,0:02:55.329 1 times 1 is 1. 0:02:55.329,0:03:01.852 Add here we get negative 6, 1 times [br]negative 6 is negative 6 and we get zero. 0:03:01.852,0:03:05.170 Now first of all, we knew that[br]was going to happen because look, 0:03:05.170,0:03:07.788 if you evaluate 1 in the [br]polynomial, we get zero. 0:03:07.788,0:03:10.708 We knew that the remainder[br]here was going to be zero. 0:03:10.708,0:03:12.758 The important part is right here. 0:03:12.758,0:03:17.223 Okay, so our polynomial [br]P of x can now be written 0:03:17.223,0:03:21.702 as x minus 1 times— [br]Read this off. 0:03:21.702,0:03:23.956 Remember, this is a [br]third-degree polynomial. 0:03:23.956,0:03:29.590 This quotient must be second degree, [br]so x squared plus x minus 6. 0:03:29.590,0:03:33.250 Now we have a quadratic [br]that we can try and factor. 0:03:33.250,0:03:35.979 I suggest you pause the [br]video and give it a shot. 0:03:35.979,0:03:38.741 It does factor; [br]I can see that it factors. 0:03:40.334,0:03:42.793 I hope you tried [br][and] paused the video. 0:03:42.793,0:03:48.029 If x minus 1 times x[br]plus 3 times x minus 2. 0:03:48.268,0:03:53.581 Now we can use the skills[br]that we learned earlier. 0:03:53.581,0:03:56.521 We could plot the zeros, [br]we know the end behavior, 0:03:56.521,0:03:59.253 and we could sketch [br]the polynomial. 0:04:00.263,0:04:04.241 Okay, one more problem[br]to finish up Section 3.3. 0:04:04.241,0:04:08.805 We're going to find a polynomial[br](fourth-degree) that has these zeros 0:04:08.805,0:04:12.795 for which the coefficient [br]x to the third is negative 6. 0:04:12.795,0:04:14.995 Okay, we’ll get [br]to this part first. 0:04:14.995,0:04:18.788 This first part is not that [br]difficult because each 0:04:18.788,0:04:23.314 zero (each x-intercept) [br]corresponds to a factor. 0:04:23.314,0:04:26.565 Let's build this polynomial; [br]we'll just call a P of x. 0:04:26.565,0:04:39.423 As x minus negative 3,[br]x minus zero, x minus 1, x minus 5. 0:04:39.423,0:04:44.373 Each one of those factors [br]corresponds to one of these zeros, 0:04:44.373,0:04:46.239 one of these x-intercepts. 0:04:46.239,0:04:47.914 We can clean it up a little. 0:04:47.914,0:04:50.707 We are going to have just x here, [br]so let's put that up front. 0:04:50.707,0:04:59.985 And then we're going to have [br]x plus 3 and x minus 1 and x minus 5. 0:05:00.531,0:05:02.458 Now this is a little tougher. 0:05:02.458,0:05:06.040 They want the coefficient of the [br]x to the third term to be negative 6. 0:05:06.040,0:05:09.267 Okay? I hope you can see[br]that if we multiply this out, 0:05:09.267,0:05:12.594 it's going to be a [br]fourth-degree polynomial. 0:05:12.594,0:05:13.964 Let's do that. 0:05:13.964,0:05:16.064 Let's multiply these two, 0:05:16.064,0:05:17.122 then these two. 0:05:17.122,0:05:19.482 And then we'll multiply the [br]two resulting polynomials. 0:05:19.482,0:05:22.138 We get x squared plus 3x. 0:05:22.138,0:05:24.602 I just multiply these two terms. 0:05:24.602,0:05:27.860 Then here we're[br]going to have x squared. 0:05:27.860,0:05:32.799 There's negative 5x minus 1x [br]is minus 6x plus 5. 0:05:33.915,0:05:35.598 Let's multiply that out. 0:05:35.598,0:05:39.967 Now I'm going to distribute [br]the two terms here 0:05:39.967,0:05:42.715 to all three terms in [br]the second polynomial. 0:05:42.715,0:05:43.677 Keep an eye on me. 0:05:43.677,0:05:52.307 X to the fourth, there's minus[br]6x to the third, plus 5x squared. 0:05:52.307,0:05:55.049 I distributed this term[br]to all three terms here. 0:05:55.049,0:05:57.495 Now let's take the 3x[br]and distribute it. 0:05:57.495,0:06:00.712 We're going to have [br]plus 3x to the third 0:06:00.712,0:06:07.403 minus 18x squared plus 15x. 0:06:11.583,0:06:14.720 Let's clean up here; [br]this is a 15x. 0:06:15.005,0:06:17.007 So, we have [br]x to the fourth. 0:06:17.007,0:06:20.101 How many x to the thirds are there? [br]Well, here's negative 6, 0:06:20.101,0:06:23.940 here's positive 3, [br]so minus 3x to the third. 0:06:23.940,0:06:30.215 Here's 5x squared minus [br]18x squared is minus 13x squared. 0:06:30.215,0:06:32.909 And then plus 15x. 0:06:34.171,0:06:37.799 Now they wanted this— [br]Let's go back to the problem. 0:06:37.799,0:06:41.308 First of all, we found a polynomial [br]of degree four that has these zeros. 0:06:41.308,0:06:44.230 This polynomial for [br]sure has these zeros. 0:06:44.230,0:06:45.907 Okay, now there's a catch. 0:06:45.907,0:06:49.082 They want the coefficient on [br]x to the third to be negative 6. 0:06:49.082,0:06:52.557 The way that we can [br]force this to be negative 6 0:06:52.557,0:07:00.242 is actually to take our polynomial P of x, [br]multiply everything by positive 2, 0:07:00.242,0:07:07.365 so 2x to the fourth, minus 6x [br]to the third, minus 26x squared— 0:07:07.365,0:07:10.687 I'm just multiplying[br]everything by two— plus 30x. 0:07:10.687,0:07:13.870 Okay, so this polynomial— [br]and I would dare you to graph this— 0:07:13.870,0:07:17.674 it's still going to have these[br]four zeros guaranteed. 0:07:17.674,0:07:22.710 But it also has this condition[br]that the x to the third term 0:07:22.710,0:07:26.791 in fact has a coefficient[br]of negative 6 as desired. 0:07:29.699,0:07:31.775 Okay, so there we go. 0:07:31.775,0:07:33.663 That's the end of section 3.3. 0:07:33.663,0:07:36.178 Finally, now we move on to 3.4. 0:07:36.178,0:07:39.210 I think I said that already in the [br]previous video, but here we go. 0:07:39.210,0:07:40.371 Bye.