WEBVTT 00:00:00.886 --> 00:00:06.296 A:middle >> Welcome to the Cypress College Math Review on Factoring Trinomials Advanced. 00:00:07.296 --> 00:00:15.176 A:middle In objective 1, you will learn how to factor quadratic type expressions using the AC method 00:00:15.956 --> 00:00:18.986 A:middle when the leading coefficient is equal to 1. 00:00:19.536 --> 00:00:30.606 A:middle In this first example, we have y to the 6th minus 14y to the 3rd plus 24. 00:00:32.276 --> 00:00:37.936 A:middle To use the AC method, we need this to be a quadratic expression. 00:00:38.386 --> 00:00:48.806 A:middle So we can do a temporary substitution where we let x equal y to the 3rd, 00:00:50.456 --> 00:00:59.726 A:middle then x squared would be y to the 3rd squared which is y to the 6th. 00:01:00.996 --> 00:01:05.956 A:middle And now we can make these substitutions in the original expression, 00:01:07.036 --> 00:01:15.776 A:middle and it becomes x squared minus 14x plus 24. 00:01:16.336 --> 00:01:25.106 A:middle Now here, we see the leading coefficient is 1, we know that we are going 00:01:25.106 --> 00:01:29.016 A:middle to make the grid here to use the AC method. 00:01:29.556 --> 00:01:39.856 A:middle And in the top here, we take the leading coefficient times the constant term which is 24. 00:01:40.476 --> 00:01:51.746 A:middle We put the B term right here, negative 14, and because the AC term is 24 00:01:52.186 --> 00:01:59.346 A:middle and the B term is negative 14 we know both factors of 24 will have to be negative. 00:02:00.676 --> 00:02:02.916 A:middle So we start walking through the factors. 00:02:03.546 --> 00:02:10.566 A:middle Negative 1 times negative 24 would add to be negative 25, 00:02:10.936 --> 00:02:16.536 A:middle which is not the negative 14 we need, but right away we see 00:02:16.536 --> 00:02:27.976 A:middle that negative 2 times negative 12 gives us the positive 24 but adds to be negative 14. 00:02:28.836 --> 00:02:37.426 A:middle And because the leading coefficient is 1 here, these immediately give us our factors, 00:02:38.196 --> 00:02:44.486 A:middle so we would get x minus 2 times x minus 12. 00:02:46.796 --> 00:02:52.916 A:middle Now, remember that this was a temporary substitution 00:02:54.186 --> 00:02:57.956 A:middle because the original expression was in terms of the y variable. 00:02:58.636 --> 00:03:05.906 A:middle So we have to go back to that and wherever we see an x, we are going to put y cubed. 00:03:06.826 --> 00:03:15.876 A:middle So this becomes y cubed minus 2 times y cubed minus 12. 00:03:18.456 --> 00:03:22.726 A:middle Now we always need to FOIL out to check our factoring. 00:03:24.076 --> 00:03:27.756 A:middle First times first gives us y to the 6th. 00:03:29.076 --> 00:03:37.956 A:middle The outer inner term, we have a negative 12y cubed and a negative 2y cubed. 00:03:38.606 --> 00:03:42.756 A:middle Those combined to give us negative 14y cubed. 00:03:43.756 --> 00:03:51.256 A:middle And then last times last gives us positive 24, so we have a correct factorization. 00:04:00.356 --> 00:04:04.946 A:middle In this example, we have to factor the quantity a minus 1 00:04:04.946 --> 00:04:11.866 A:middle to the 10th plus 8 times the quantity a minus 1 to the 5th plus 15. 00:04:12.456 --> 00:04:19.496 A:middle Now this trinomial can be turned into a quadratic type trinomial 00:04:20.076 --> 00:04:26.556 A:middle and the way you recognize that is because the highest degree is always double 00:04:27.006 --> 00:04:29.086 A:middle of the next highest degree. 00:04:29.776 --> 00:04:33.266 A:middle When that happens, this can be a quadratic type. 00:04:33.976 --> 00:04:44.136 A:middle So for example, we would let x equal the quantity a minus 1 to the 5th power. 00:04:45.346 --> 00:04:55.816 A:middle And then the x squared term would be that quantity a minus 1 to the 5th squared 00:04:56.736 --> 00:05:05.726 A:middle which is equal to a minus 1 to the 10th which is exactly what our highest degree is here. 00:05:08.216 --> 00:05:11.046 A:middle Now if we make these substitutions, 00:05:11.386 --> 00:05:20.596 A:middle then this trinomial becomes x squared plus 8 times x plus 15. 00:05:21.486 --> 00:05:24.776 A:middle And now this looks like a very simple thing to factor. 00:05:25.336 --> 00:05:30.416 A:middle So using the AC method, we make the X grid. 00:05:31.816 --> 00:05:36.216 A:middle Our leading coefficient times the constant term is 15. 00:05:37.016 --> 00:05:39.716 A:middle The B term is 8. 00:05:40.356 --> 00:05:45.746 A:middle All the terms are positive so we know both factors will be positive. 00:05:46.626 --> 00:05:53.576 A:middle And we know 1 and 15 would add to be 16, so that's not it because we need 8. 00:05:54.486 --> 00:05:59.006 A:middle And we see quickly here that the factors need to be 3 00:05:59.006 --> 00:06:04.046 A:middle and 5 so that they will multiply to be 15 but add to be 8. 00:06:05.496 --> 00:06:08.256 A:middle And because the leading coefficient is a 1, 00:06:08.796 --> 00:06:18.286 A:middle these two give us the factorization immediately, x plus 3 times x plus 5. 00:06:18.896 --> 00:06:25.906 A:middle And remember, this is a temporary substitution that we made. 00:06:26.706 --> 00:06:32.306 A:middle We have to go back and put it in the same form as the original expression. 00:06:33.476 --> 00:06:39.756 A:middle Wherever we see an x, we're going to replace it with a minus 1 to the 5th power. 00:06:40.346 --> 00:06:49.336 A:middle So in this first factor, then we would get a minus 1 to the 5th plus 3. 00:06:49.906 --> 00:06:59.946 A:middle And the second factor would become quantity a minus 1 to the 5th plus 5. 00:07:03.136 --> 00:07:07.056 A:middle Now, we always want to FOIL to check our answer. 00:07:08.326 --> 00:07:13.276 A:middle First times first will give us a minus 1 to the 10th. 00:07:13.826 --> 00:07:21.056 A:middle For the outer inner term, we're going to get 5 times a minus 1 00:07:21.056 --> 00:07:30.416 A:middle to the 5th plus 3 times a minus 1 to the 5th and that gives us our 8 times a minus 1 to the 5th. 00:07:31.306 --> 00:07:39.146 A:middle And then last times last will give us 15 so we have a correct factorization. 00:07:50.356 --> 00:07:58.996 A:middle In this example, we will factor h to the 8th power minus 4 times h to the 4th, 00:07:59.496 --> 00:08:04.306 A:middle k to the 3rd minus 12k to the 6th. 00:08:04.826 --> 00:08:13.666 A:middle This is another one that we can turn into a quadratic type expression. 00:08:15.256 --> 00:08:23.786 A:middle Notice that this exponent of 8 is double the exponent of 4 for the h variable. 00:08:24.276 --> 00:08:33.076 A:middle And then the k variable, the exponent of 6, is double the exponent of 3. 00:08:35.156 --> 00:08:41.636 A:middle So for this problem, we are going to end up having two variables that we substitute. 00:08:42.136 --> 00:08:55.496 A:middle We could let a equal h to the 4th so that a squared would then be h 00:08:55.496 --> 00:09:01.506 A:middle to the 4th squared which is h to the 8th. 00:09:03.336 --> 00:09:15.696 A:middle We will also let b equal k to the 3rd so that b squared is equal to k 00:09:15.696 --> 00:09:20.776 A:middle to the 3rd squared which is k to the 6th. 00:09:22.876 --> 00:09:36.486 A:middle Now, when we do these substitutions, we replace h to the 8th with a squared minus 4 times h 00:09:36.486 --> 00:09:48.946 A:middle to the 4th gets replaced with a, k to the 3rd gets replaced with b minus 12, 00:09:49.356 --> 00:09:54.926 A:middle and then k to the 6th will get replaced with b squared. 00:09:55.496 --> 00:10:06.286 A:middle We see the leading coefficient is 1 so we can use the short form of the AC factoring. 00:10:08.236 --> 00:10:15.166 A:middle The leading coefficient times the last coefficient is negative 12. 00:10:16.416 --> 00:10:20.706 A:middle The middle coefficient is negative 4. 00:10:22.256 --> 00:10:28.516 A:middle So we need two numbers that multiply to negative 12 but add to be negative 4. 00:10:29.746 --> 00:10:34.706 A:middle We could try 1 and negative 12 but that adds to negative 11. 00:10:35.896 --> 00:10:43.886 A:middle And next, 2 and negative 6 multiplied to be negative 12 but add to be negative 4. 00:10:45.176 --> 00:10:51.766 A:middle Because the leading coefficient is 1, these immediately give us the factors, 00:10:52.376 --> 00:10:56.336 A:middle but we have to be careful here because we have two variables going on. 00:10:57.576 --> 00:11:08.706 A:middle So this means the factorization will be a plus 2 times b. Because remember, 00:11:08.706 --> 00:11:11.576 A:middle we have this b variable going on. 00:11:12.686 --> 00:11:23.126 A:middle And then the second factor will be a minus 6 times b. Now we have to substitute back 00:11:23.936 --> 00:11:27.256 A:middle to the variables that were in the original expression. 00:11:28.716 --> 00:11:34.826 A:middle Wherever we see an a, we're going to replace that with h to the 4th. 00:11:37.436 --> 00:11:43.136 A:middle Wherever we see a b, we're going to replace that with k to the 3rd. 00:11:45.796 --> 00:11:54.446 A:middle The second factor then will become h to the 4th minus 6k to the 3rd. 00:11:55.106 --> 00:12:00.376 A:middle Now we're ready to FOIL to check our answer. 00:12:01.336 --> 00:12:08.046 A:middle First times first will give us h to the 8th. 00:12:08.296 --> 00:12:14.806 A:middle The outer term is negative 6 h to the 4th k cubed. 00:12:15.376 --> 00:12:21.056 A:middle The inner term is positive 2 h to the 4th k cubed. 00:12:21.766 --> 00:12:27.626 A:middle So, negative 6 plus a positive 2 gives us that negative 4 h 00:12:27.626 --> 00:12:31.296 A:middle to the 4th k cubed that we need in the middle term. 00:12:32.306 --> 00:12:39.016 A:middle And then last times last gives us our negative 12k to the 6th. 00:12:39.526 --> 00:12:42.966 A:middle So we have a correct factorization. 00:12:51.516 --> 00:12:54.026 A:middle It's time to check your understanding. 00:12:54.636 --> 00:12:59.086 A:middle So pause the video to try this one on your own, then restart the video 00:12:59.376 --> 00:13:00.996 A:middle when you're ready to check your answer. 00:13:05.916 --> 00:13:09.406 A:middle In this example, we have quantity x plus y 00:13:09.406 --> 00:13:17.696 A:middle to the 4th power plus 35 times the quantity x plus y squared minus 36. 00:13:19.156 --> 00:13:25.386 A:middle You see right away that this can be turned into a quadratic type trinomial 00:13:26.106 --> 00:13:31.456 A:middle because the highest degree is double the next highest degree. 00:13:32.656 --> 00:13:41.156 A:middle So we can do a substitution where we let a equal x plus y squared, 00:13:42.416 --> 00:13:48.096 A:middle then a squared would be the quantity x plus y squared, 00:13:48.716 --> 00:13:55.396 A:middle squared which would turn it into x plus y to the 4th power. 00:13:55.916 --> 00:14:05.726 A:middle When we do these substitutions, x plus y to the 4th power, gets replaced with a squared. 00:14:06.856 --> 00:14:18.136 A:middle We have plus 35 times the x plus y squared gets replaced with a and then minus 36. 00:14:19.736 --> 00:14:25.986 A:middle This is a simple quadratic trinomial that we can factor with the AC method 00:14:26.066 --> 00:14:28.386 A:middle where the leading coefficient is 1. 00:14:29.346 --> 00:14:36.836 A:middle We put the constant term here of negative 36, the B term, positive 35, 00:14:37.436 --> 00:14:47.896 A:middle and we get right away the factorization is going to be a minus 1 and a positive 36 00:14:48.716 --> 00:14:54.496 A:middle because they multiply to be negative 36 but add to be 35. 00:14:55.746 --> 00:15:01.886 A:middle Because the leading coefficient is a 1, these numbers immediately give us the factorization 00:15:03.056 --> 00:15:10.506 A:middle and we get a minus 1 times a plus 36. 00:15:11.036 --> 00:15:17.046 A:middle We have to substitute back to the original variables. 00:15:18.896 --> 00:15:26.546 A:middle Everywhere we see an a, we are going to replace it with an x plus y quantity squared. 00:15:27.186 --> 00:15:36.346 A:middle So in this first factor, we are going to get x plus y quantity squared minus 1, 00:15:36.876 --> 00:15:46.146 A:middle and then the second factor we are going to get x plus y quantity squared plus 36. 00:15:48.736 --> 00:15:52.196 A:middle Now let's use FOIL to check our answer. 00:15:53.226 --> 00:15:58.496 A:middle First times first gives us quantity x plus y to the 4th, 00:15:59.686 --> 00:16:04.786 A:middle the outer term is a positive 36 times the quantity, 00:16:04.846 --> 00:16:10.446 A:middle the inner term is a minus 1 times the quantity so the positive 36 00:16:10.516 --> 00:16:17.776 A:middle and the minus 1 give us the positive 35 times that quantity of x plus y squared. 00:16:18.756 --> 00:16:23.076 A:middle And then last times last we get negative 36. 00:16:24.526 --> 00:16:27.016 A:middle So we have a correct factorization. 00:16:28.126 --> 00:16:32.096 A:middle However, on this problem, we're not quite done yet. 00:16:33.336 --> 00:16:40.606 A:middle Did any of you notice that this first factor here is the difference of two squares? 00:16:41.476 --> 00:16:44.966 A:middle Now, this didn't come up on any of our previous examples. 00:16:45.316 --> 00:16:50.126 A:middle So if you didn't notice that, don't feel bad, but this is an additional step 00:16:50.476 --> 00:16:54.156 A:middle because this is not quite factored all the way. 00:16:55.436 --> 00:17:11.956 A:middle This first one is going to factor as x plus y plus 1 times x plus y minus 1 because this is 00:17:11.956 --> 00:17:18.256 A:middle like having a squared minus b squared and we get a plus b, 00:17:18.726 --> 00:17:25.726 A:middle a minus b. The second factor cannot be factored anymore, so we bring it right 00:17:25.766 --> 00:17:33.186 A:middle down as x plus y quantity squared plus 36. 00:17:35.196 --> 00:17:38.796 A:middle Now, we have a complete factorization. 00:17:40.156 --> 00:17:43.766 A:middle And if you saw this one and did that factoring, great job. 00:17:44.006 --> 00:17:46.156 A:middle If you didn't see it, that's OK. 00:17:46.336 --> 00:17:51.456 A:middle Just a reminder that when you end up with the difference of two squares, 00:17:51.796 --> 00:17:54.376 A:middle you do have to factor it further. 00:18:02.096 --> 00:18:05.626 A:middle Below are some practice problems you can try on your own. 00:18:06.486 --> 00:18:12.336 A:middle You may either pause the video to work on them now or write them down to work on them later. 00:18:13.416 --> 00:18:16.816 A:middle After a few seconds, the answers will be revealed. 00:18:30.156 --> 00:18:36.786 A:middle In objective 2, you will learn how to factor quadratic type expressions using the AC method 00:18:37.416 --> 00:18:40.306 A:middle when the leading coefficient does not equal 1. 00:18:40.856 --> 00:18:50.716 A:middle In this first example, we have 12x to the 6th plus 8x to the 3rd minus 7. 00:18:51.216 --> 00:18:57.436 A:middle And you can see that it's going to be a quadratic type expression 00:18:57.866 --> 00:19:03.776 A:middle because the highest exponent 6 is double the middle exponent of 3. 00:19:04.426 --> 00:19:14.166 A:middle Now we can do that temporary substitution where we let a equal x to the 3rd, 00:19:14.686 --> 00:19:25.306 A:middle then we know a squared would be x to the 3rd squared which is x to the 6th. 00:19:26.356 --> 00:19:35.326 A:middle So we can rewrite this expression as 12a squared plus 8a minus 7. 00:19:35.866 --> 00:19:39.826 A:middle Let's make our X grid. 00:19:41.326 --> 00:19:47.596 A:middle We know the AC term, 12 times negative 7, is negative 84. 00:19:48.776 --> 00:19:56.556 A:middle The B term is positive 8 because the factors have to multiply 00:19:56.556 --> 00:20:00.596 A:middle to be a negative but add to be a positive. 00:20:01.136 --> 00:20:05.876 A:middle We know that the larger magnitude factor will be the one that we make positive. 00:20:07.026 --> 00:20:12.826 A:middle So let's walk through the factors of negative 84 and see which ones work. 00:20:13.746 --> 00:20:26.936 A:middle If we do negative 1 times 84, we know that adds to 83 which is not 8, negative 2 times 42 adds 00:20:26.936 --> 00:20:37.056 A:middle to be 40 which is not 8, negative 3 times 28 adds to 25 which doesn't work, 00:20:38.386 --> 00:20:46.916 A:middle negative 4 times 21 adds to be 17, again, doesn't work, but we see when we get 00:20:46.946 --> 00:20:55.046 A:middle to negative 6 times a positive 14, those two multiply together to be negative 84 00:20:55.236 --> 00:20:58.356 A:middle but they do add to be positive 8. 00:20:59.006 --> 00:21:08.556 A:middle Now when the leading coefficient does not equal 1, these two numbers tell us how to break 00:21:08.556 --> 00:21:15.256 A:middle up that middle term into minus 6a plus14a. 00:21:15.416 --> 00:21:25.946 A:middle We know the leading term comes right down 12a squared, the third term comes right 00:21:25.946 --> 00:21:33.046 A:middle down minus 7, and now we're ready to factor two by two. 00:21:33.536 --> 00:21:44.596 A:middle From the first two terms, we can factor out 6a and we would have left 2a minus 1. 00:21:45.346 --> 00:21:46.336 A:middle Let's check that. 00:21:46.826 --> 00:21:54.406 A:middle Six a times 2a is 12a squared, 6a times a negative 1 is negative 6a. 00:21:56.256 --> 00:22:00.776 A:middle Whatever math symbol is here, in this case a plus sign, comes right down. 00:22:02.066 --> 00:22:09.356 A:middle And from these two, we can factor out a 7 and we have left 2a minus 1. 00:22:10.576 --> 00:22:18.846 A:middle Once again we check, 7 times 2a is 14a, 7 times negative 1 is negative 7. 00:22:20.596 --> 00:22:32.146 A:middle Now from these two terms, we can factor out 2a minus 1 and we have left 6a plus 7. 00:22:34.036 --> 00:22:43.186 A:middle But remember, this all was a temporary substitution and we have to convert back 00:22:43.856 --> 00:22:46.416 A:middle to the variables from the original expression. 00:22:47.896 --> 00:22:52.976 A:middle Everywhere we see an a we will instead put x cubed, 00:22:54.106 --> 00:23:04.296 A:middle so this becomes 2x cubed minus 1 times 6x cubed plus 7, 00:23:04.826 --> 00:23:09.546 A:middle and we are ready to FOIL to check our answer. 00:23:10.876 --> 00:23:18.126 A:middle First times first we have 2x cubed times 6x cubed is 12x to the 6th. 00:23:19.296 --> 00:23:22.376 A:middle When we combine the outer and inner terms, 00:23:22.966 --> 00:23:29.926 A:middle we'll get 14x cubed minus 6x cubed gives us the 8x cubed. 00:23:30.496 --> 00:23:39.636 A:middle And then last times last gives us a negative 7, so we have a correct factorization. 00:23:48.176 --> 00:23:53.346 A:middle In this example we have to factor 3 times the quantity y plus 2 00:23:53.346 --> 00:24:01.276 A:middle to the 4th power plus 13 times the quantity y plus 2 to the 2nd power minus 10. 00:24:02.246 --> 00:24:10.956 A:middle Once again, we recognize this as a quadratic type expression, because 4 is double of 2. 00:24:11.476 --> 00:24:17.066 A:middle We're going to use the technique where we do a temporary substitution. 00:24:17.656 --> 00:24:25.826 A:middle So let's let a equal y plus 2 quantity squared. 00:24:27.176 --> 00:24:34.596 A:middle Then a squared would be that same quantity y plus 2 quantity squared, 00:24:35.776 --> 00:24:41.936 A:middle squared which is the quantity y plus 2 to the 4th. 00:24:45.136 --> 00:24:47.716 A:middle Now we can do those substitutions 00:24:47.996 --> 00:24:58.016 A:middle and temporarily the expression will become 3a squared plus 13a minus 10. 00:25:00.216 --> 00:25:05.216 A:middle This is now a simple AC method type factoring problem. 00:25:05.376 --> 00:25:07.316 A:middle So we make our X grid. 00:25:09.456 --> 00:25:13.416 A:middle The leading coefficient times the constant term, 00:25:13.566 --> 00:25:21.206 A:middle 3 times negative 10 is negative 30, the B term is positive 13. 00:25:22.366 --> 00:25:27.836 A:middle They have to multiply to be negative but add to be positive so we know we're going 00:25:27.836 --> 00:25:33.246 A:middle to have a negative times a positive and we need the larger magnitude factor to be positive. 00:25:35.286 --> 00:25:41.846 A:middle So negative 1 times 30 adds to be 29, which is not 13, 00:25:42.906 --> 00:25:48.316 A:middle but already negative 2 times a positive 15 multiplies 00:25:48.316 --> 00:25:52.016 A:middle to be our negative 30 but adds to be positive 13. 00:25:54.236 --> 00:26:01.806 A:middle These numbers when the leading coefficient is not 1, tell us how to break 00:26:01.806 --> 00:26:08.836 A:middle up the B term into negative 2a plus 15a. 00:26:09.406 --> 00:26:19.016 A:middle We know the leading term comes right down 3a squared, the constant term comes straight 00:26:19.046 --> 00:26:25.526 A:middle down minus 10, and now we are ready to factor two by two. 00:26:26.026 --> 00:26:36.616 A:middle From these first two, the only thing we can factor out is a and we would have 3a minus 2. 00:26:37.476 --> 00:26:47.826 A:middle Always check a times 3a is 3a squared, a times a negative 2 is negative 2a. 00:26:48.046 --> 00:26:53.006 A:middle Whatever math symbol is here, in this case a plus, we bring it down. 00:26:53.596 --> 00:26:56.916 A:middle From these two, we can factor out a 5. 00:26:57.976 --> 00:27:01.636 A:middle And we would have 3a minus 2. 00:27:02.256 --> 00:27:12.116 A:middle We always check, 5 times 3a is 15a, 5 times negative 2 is negative 10. 00:27:12.336 --> 00:27:23.986 A:middle Now from these two terms, we can factor out a 3a minus 2 and we have left a plus 5. 00:27:28.436 --> 00:27:34.166 A:middle Now remember, this all was a temporary substitution. 00:27:35.106 --> 00:27:39.866 A:middle So we have to convert back to the variables from the original expression 00:27:40.596 --> 00:27:45.666 A:middle and everywhere we see an a we are going to substitute it 00:27:46.076 --> 00:27:48.966 A:middle with the quantity y plus 2 squared. 00:27:49.966 --> 00:28:00.786 A:middle So this first factor becomes 3 times the quantity y plus 2 squared minus 2. 00:28:01.316 --> 00:28:12.846 A:middle And the second factor becomes quantity y plus 2 squared plus 5. 00:28:13.336 --> 00:28:19.326 A:middle We are ready to FOIL to check our answer. 00:28:20.346 --> 00:28:26.366 A:middle First times first gives us 3y plus 2 to the 4th. 00:28:27.636 --> 00:28:30.446 A:middle Combining the outer and inner terms, 00:28:30.916 --> 00:28:39.776 A:middle we get 15 times the quantity y plus 2 squared minus 2 times the quantity y plus 2 squared, 00:28:39.866 --> 00:28:49.956 A:middle which is 13 times the quantity y plus 2 squared, and then last times last gives us the minus 10. 00:28:51.336 --> 00:28:53.976 A:middle So we have a correct factorization. 00:29:04.716 --> 00:29:13.886 A:middle In this example, we have to factor 8x to the 2/3 minus 26x to the 1/3 plus 15. 00:29:14.446 --> 00:29:21.466 A:middle Once again, we recognize that this is also a quadratic type expression 00:29:22.106 --> 00:29:26.566 A:middle because 2/3 is double of 1/3. 00:29:27.066 --> 00:29:37.866 A:middle Using the technique where we do the temporary substitution, let's let a equal x to the 1/3, 00:29:39.276 --> 00:29:47.566 A:middle then a squared would be x to the 1/3 quantity squared. 00:29:48.146 --> 00:29:50.936 A:middle And remember, a power to a power you multiply. 00:29:51.396 --> 00:29:54.896 A:middle So this becomes x to the 2/3. 00:29:57.056 --> 00:30:10.216 A:middle Now we're ready to do the substitution and the expression becomes 8a squared minus 26a plus 15. 00:30:11.806 --> 00:30:15.416 A:middle Using the AC method, we make the X grid. 00:30:16.716 --> 00:30:29.756 A:middle The leading coefficient times the constant term 8 times 15 is 120 and the B term is negative 26. 00:30:31.096 --> 00:30:37.146 A:middle Our two factors have to multiply to be a positive but add to be a negative, 00:30:37.376 --> 00:30:40.146 A:middle so we know that they're both going to be negative. 00:30:41.466 --> 00:30:48.226 A:middle So let's walk through the factors of 120 until we find the ones that work. 00:30:49.976 --> 00:30:58.466 A:middle Negative 1 times 120 would add to be negative 121, that doesn't work. 00:30:59.676 --> 00:31:06.576 A:middle Negative 2 times negative 60 adds to be negative 62 which does not work. 00:31:08.046 --> 00:31:16.576 A:middle Negative 3 times negative 40 adds to be negative 43, getting close but still doesn't work. 00:31:18.096 --> 00:31:24.196 A:middle Negative 4 times 30 adds to be negative 34. 00:31:25.816 --> 00:31:33.376 A:middle Negative 5 times negative 24 adds to be negative 29, we're getting really close. 00:31:34.536 --> 00:31:44.146 A:middle And we can see that negative 6 times negative 20 multiplied to be 120 add to be negative 26. 00:31:45.246 --> 00:31:50.536 A:middle Now, if you can see these factors right away, then of course you just want to write them down 00:31:50.536 --> 00:31:53.936 A:middle and you don't need to go through this whole list in order to get there. 00:31:54.626 --> 00:31:59.446 A:middle However, I've found that students tend to do better when they go 00:31:59.446 --> 00:32:05.426 A:middle through an organized list starting from 1 and working their way up then they don't end 00:32:05.426 --> 00:32:07.836 A:middle up missing any factors that would work. 00:32:08.476 --> 00:32:17.116 A:middle Now because the leading coefficient is 8 and not 1, these two numbers tell us how to break 00:32:17.116 --> 00:32:31.196 A:middle up the middle term into negative 6a minus 20a, the leading term comes right down 8a squared, 00:32:31.926 --> 00:32:36.256 A:middle and the constant term comes right down plus 15. 00:32:38.056 --> 00:32:41.166 A:middle Now we're ready to factor two by two. 00:32:41.766 --> 00:32:51.706 A:middle So from these first two, we can factor out a 2a and we would have left 4a minus 3. 00:32:52.526 --> 00:33:01.246 A:middle Let's check it, 2a times 4a is 8a squared, 2a times a negative 3, negative 6a. 00:33:02.516 --> 00:33:07.816 A:middle Whatever the math symbol is here, in this case a minus sign, we bring that down. 00:33:08.806 --> 00:33:12.526 A:middle And then from these two, we can factor out a 5. 00:33:13.846 --> 00:33:19.846 A:middle And when we do that division, we would have left 4a minus 3. 00:33:20.536 --> 00:33:21.506 A:middle Let's check it. 00:33:22.466 --> 00:33:26.186 A:middle Negative 5 times 4a, negative 20a. 00:33:26.936 --> 00:33:30.926 A:middle Negative 5 times negative 3, positive 15. 00:33:31.486 --> 00:33:41.366 A:middle Now the greatest common factor of these two terms is 4a minus 3 00:33:42.676 --> 00:33:46.826 A:middle and we have left over 2a minus 5. 00:33:50.036 --> 00:33:57.776 A:middle But remember, we did a temporary substitution so we will have to go back 00:33:57.826 --> 00:34:00.456 A:middle to the variables in the original expression. 00:34:01.596 --> 00:34:07.256 A:middle Everywhere we see an a, we will put instead x to the 1/3 power. 00:34:08.226 --> 00:34:22.156 A:middle So this becomes 4 times x to the 1/3 minus 3 times the quantity 2 times x to the 1/3 minus 5. 00:34:22.656 --> 00:34:27.456 A:middle And we're ready to FOIL to check our answer. 00:34:28.806 --> 00:34:37.696 A:middle First times first gives us 8x to the 2/3 because x to the 1/3 times x 00:34:37.696 --> 00:34:43.066 A:middle to the 1/3 means you add those together, 1/3 plus 1/3 is 2/3. 00:34:45.016 --> 00:34:57.706 A:middle The outer inner terms become minus 20x to the 1/3 minus 6x to the 1/3 which is minus 26x 00:34:58.056 --> 00:35:05.096 A:middle to the 1/3, and then a negative 3 times a negative 5 is a positive 15. 00:35:05.726 --> 00:35:08.466 A:middle So we have a correct factorization. 00:35:19.766 --> 00:35:22.126 A:middle It's time to check your understanding. 00:35:22.536 --> 00:35:26.976 A:middle So pause the video to try this one your own, then restart the video 00:35:27.176 --> 00:35:28.736 A:middle when you're ready to check your answer. 00:35:34.476 --> 00:35:38.206 A:middle In this example, we have to factor 20x 00:35:38.206 --> 00:35:46.666 A:middle to the 4th plus 36x squared y to the 8th plus 9y to the 16th. 00:35:47.156 --> 00:35:56.736 A:middle Now this can be turned into a quadratic type expression because we notice that the exponent 00:35:56.736 --> 00:36:00.296 A:middle of 4 is twice the exponent of 2, 00:36:01.006 --> 00:36:07.996 A:middle and the exponent of the y variable 16 is twice the exponent 00:36:07.996 --> 00:36:10.326 A:middle of the middle y variable which is 8. 00:36:11.076 --> 00:36:14.476 A:middle So we are going to do a double substitution here. 00:36:16.236 --> 00:36:27.786 A:middle So let's let a equal x squared so that a squared would be x squared, 00:36:27.866 --> 00:36:31.706 A:middle squared which of course is x to the 4th. 00:36:32.276 --> 00:36:45.086 A:middle We are also going to let b equal y to the 8th so that b squared would then be y 00:36:45.086 --> 00:36:49.816 A:middle to the 8th squared, which is y to the 16th. 00:36:52.616 --> 00:36:57.516 A:middle Now let's come over to the original expression and make those substitutions. 00:36:58.416 --> 00:37:16.376 A:middle Then this would become 20 times a squared plus 36 times a times b plus 9 times b squared. 00:37:17.706 --> 00:37:21.276 A:middle Using the AC method, we make the X grid, 00:37:21.896 --> 00:37:30.116 A:middle and the leading coefficient times the last coefficient, 20 times 9, is 180. 00:37:31.606 --> 00:37:35.146 A:middle The B term is a positive 36. 00:37:36.976 --> 00:37:42.316 A:middle So we know that both factors will be positive because when we multiply and we get a positive, 00:37:42.316 --> 00:37:44.096 A:middle when we add them we get a positive. 00:37:44.936 --> 00:37:51.256 A:middle So let's work our way through the factors of 180 and find the ones that add to be 36. 00:37:52.406 --> 00:37:58.556 A:middle We know 1 and 180 add to be 181, which doesn't work. 00:37:59.746 --> 00:38:05.366 A:middle We know 2 times 90 adds to be 92, doesn't work. 00:38:06.756 --> 00:38:13.886 A:middle Three times 60 adds to be 63, does not match that 36. 00:38:15.026 --> 00:38:20.956 A:middle Four times 45 gives us 49, still doesn't work. 00:38:22.656 --> 00:38:29.856 A:middle Five times 36 adds up to be 41, getting closer. 00:38:30.716 --> 00:38:39.336 A:middle And then we see that 6 times 30 will multiply to be 180 but add to be 36. 00:38:40.916 --> 00:38:48.056 A:middle Because the leading coefficient is not 1, these two numbers tell us how to break 00:38:48.056 --> 00:38:56.056 A:middle up the middle term into 6ab plus 30ab. 00:38:57.106 --> 00:39:01.296 A:middle And of course we're adding that 6. 00:39:03.196 --> 00:39:09.466 A:middle The leading term comes straight down as 20a squared, 00:39:10.736 --> 00:39:16.476 A:middle and the last term come straight down as plus 9b squared. 00:39:19.036 --> 00:39:22.146 A:middle Now we're ready to factor two by two. 00:39:22.686 --> 00:39:28.726 A:middle From these first two we can factor out a 2a. 00:39:29.336 --> 00:39:45.136 A:middle And when we do the division, 20a squared divided by 2a is 10a plus 6ab divided by 2a is 3b. 00:39:47.076 --> 00:39:48.086 A:middle Always check it. 00:39:48.796 --> 00:39:51.956 A:middle Two a times 10a, 20a squared. 00:39:52.596 --> 00:39:55.796 A:middle Two a times 3b, 6ab. 00:39:57.456 --> 00:40:02.476 A:middle Whatever math symbol is here, which in this case is a plus, we bring it straight down. 00:40:03.796 --> 00:40:08.236 A:middle Now from these two, we can factor out a 3b. 00:40:08.836 --> 00:40:16.646 A:middle If we do the division, 30ab divided by 3b is going 00:40:16.646 --> 00:40:26.876 A:middle to be 10a plus 9b squared divided 3b will be 3b. 00:40:28.146 --> 00:40:39.086 A:middle Let's check it, 3b times 10a is 30ab, 3b times 3b is 9b squared. 00:40:42.356 --> 00:40:51.426 A:middle Now from these two terms, the greatest common factor is 10a plus 3b, 00:40:52.006 --> 00:40:58.586 A:middle and we have left 2a plus 3b. 00:40:59.196 --> 00:41:06.446 A:middle And this was our temporary substitution. 00:41:09.776 --> 00:41:14.426 A:middle So we have to return to the variables in the original expression. 00:41:15.736 --> 00:41:20.416 A:middle Everywhere we see an a, we will trade it for an x squared, 00:41:21.026 --> 00:41:32.266 A:middle so this first factor becomes 10x squared plus, everywhere we see a b, we are going to trade it 00:41:32.266 --> 00:41:39.196 A:middle for y to the 8th so this second term in here becomes 3y to the 8th. 00:41:39.676 --> 00:41:50.836 A:middle In the second factor here, we've got 2 times x squared which substitutes for the a, 00:41:51.766 --> 00:41:57.726 A:middle plus 3 times y to the 8th which substitutes for the b, 00:41:59.246 --> 00:42:01.896 A:middle and we're ready to FOIL to check our answer. 00:42:03.336 --> 00:42:08.756 A:middle First times first we see we will get 20x to the 4th. 00:42:09.326 --> 00:42:19.766 A:middle When we combine the outer term with the inner term, we get 30x squared y 00:42:19.766 --> 00:42:29.746 A:middle to the 8th plus 6x squared y to the 8th which add to be 36x squared y to the 8th. 00:42:30.276 --> 00:42:36.616 A:middle And then last times last we get 9y to the 16th. 00:42:37.766 --> 00:42:40.556 A:middle So we have a correct factorization. 00:42:47.656 --> 00:42:51.286 A:middle Below are some practice problems you can try on your own. 00:42:52.146 --> 00:42:58.006 A:middle You may either pause the video to work on them now or write them down to work on them later. 00:42:59.086 --> 00:43:02.486 A:middle After a few seconds, the answers will be revealed. 00:43:15.296 --> 00:43:21.286 A:middle In factoring trinomials advanced, objective 3, you will learn how 00:43:21.286 --> 00:43:25.916 A:middle to factor quadratic type expressions using trial and error. 00:43:26.496 --> 00:43:36.896 A:middle In this example, we have 4x to the 1/2 minus 13x to the 1/4 minus 12. 00:43:38.836 --> 00:43:49.106 A:middle We recognize this as a quadratic type expression because the highest degree 1/2 is double 1/4. 00:43:49.606 --> 00:43:58.706 A:middle So we will do that temporary substitution where we let a equal x to the 1/4, 00:43:59.326 --> 00:44:15.816 A:middle a squared will then equal x to the 1/4 squared which is x to the 2/4 which is x to the 1/2. 00:44:17.246 --> 00:44:19.416 A:middle Now let's make those substitutions 00:44:19.836 --> 00:44:30.846 A:middle and our expression becomes 4 times a squared minus 13 times a minus 12. 00:44:31.326 --> 00:44:37.236 A:middle And now for this, we're going to factor using trial and error. 00:44:37.846 --> 00:44:42.756 A:middle So let's look at the factors of 4. 00:44:43.246 --> 00:44:47.566 A:middle It could be 1 times 4 or 2 times 2. 00:44:48.146 --> 00:44:58.636 A:middle For the factors of 12, we have 1 times 12, 2 times 6, or 3 times 4. 00:44:59.236 --> 00:45:03.616 A:middle We have to pick one to start. 00:45:03.996 --> 00:45:10.626 A:middle So maybe to get 4a squared, maybe we'll try doing the 2 times 2. 00:45:11.216 --> 00:45:14.816 A:middle So we'll try 2a times 2a. 00:45:17.116 --> 00:45:20.766 A:middle And for 12, we have three choices. 00:45:21.596 --> 00:45:25.316 A:middle Maybe we would try 3 times 4. 00:45:26.276 --> 00:45:30.266 A:middle Now we know one of them is going to be positive and one is going to be negative 00:45:30.396 --> 00:45:32.436 A:middle because of this negative 12 here. 00:45:34.036 --> 00:45:38.896 A:middle Before we decide which one should be negative, let's take a look at the cross products. 00:45:40.576 --> 00:45:43.426 A:middle We get 6a here. 00:45:44.526 --> 00:45:47.036 A:middle We get 8a here. 00:45:47.626 --> 00:45:53.686 A:middle Now because the middle term is negative, we know we're always going 00:45:53.686 --> 00:46:00.456 A:middle to want the larger cross product to be negative, which would make this 4 negative. 00:46:01.446 --> 00:46:10.036 A:middle However, when we add these up, we get a negative 2a where we needed negative 13a. 00:46:10.666 --> 00:46:19.326 A:middle So maybe we will try trading out the 3 and the 4 for a 2 and a 6. 00:46:20.376 --> 00:46:29.186 A:middle So we'll say 2a times 2a and we'll put in a 2 and a 6. 00:46:30.146 --> 00:46:37.946 A:middle The first cross product is 4a, the second cross product is 12a, 00:46:38.636 --> 00:46:43.566 A:middle and we always want the bigger one to be negative since the middle term is negative. 00:46:44.356 --> 00:46:46.566 A:middle That would make that a negative 6. 00:46:47.866 --> 00:46:49.546 A:middle This is a negative 12. 00:46:49.646 --> 00:46:53.416 A:middle When we add these two we get negative 8a. 00:46:54.186 --> 00:46:58.106 A:middle That's not the negative 13a that we need. 00:46:59.216 --> 00:47:05.216 A:middle So maybe instead of using 2 times 2, we'll try doing 1 times 4. 00:47:05.796 --> 00:47:19.006 A:middle And if we do a times 4a, maybe we would try the 3 and the 4 here. 00:47:20.326 --> 00:47:24.006 A:middle This first cross product is 12a. 00:47:24.786 --> 00:47:27.996 A:middle The second cross product is 4a. 00:47:28.726 --> 00:47:35.676 A:middle If we make the bigger one negative, we would get negative 8a, which doesn't work. 00:47:36.916 --> 00:47:39.296 A:middle We could try reversing these. 00:47:39.916 --> 00:47:52.896 A:middle If we do a and 4a and switch the 4 and 3, this first cross product is 16a, 00:47:53.396 --> 00:47:59.986 A:middle the second cross product is 3a, and we want the larger one to be negative. 00:48:00.676 --> 00:48:07.086 A:middle Now when we add these two together, we get negative 13a 00:48:07.616 --> 00:48:10.236 A:middle which is exactly what we were looking for here. 00:48:10.846 --> 00:48:29.316 A:middle So this now tells us that our factors are going to be a minus 4 times 4a plus 3. 00:48:31.576 --> 00:48:41.786 A:middle Now remember, we did this temporary substitution here, so we need to go back to the variables 00:48:41.786 --> 00:48:46.926 A:middle from the original expression and everywhere that we see a, 00:48:46.926 --> 00:48:51.936 A:middle we are going to substitute with x to the 1/4. 00:48:52.896 --> 00:49:00.976 A:middle So this becomes x to the 1/4 minus 4 times 4. 00:49:02.176 --> 00:49:07.116 A:middle And this a gets substituted with x to the 1/4 again plus 3. 00:49:07.736 --> 00:49:12.266 A:middle It's to FOIL to check our answer. 00:49:13.796 --> 00:49:25.306 A:middle First times first we get x to the 4th times 4x to the 4th which is 4x to the 1/2. 00:49:25.526 --> 00:49:28.596 A:middle Next we look at the outer and inner terms. 00:49:29.426 --> 00:49:36.566 A:middle The outer is 3x to the 4th, the inner is negative 16x to the 4th. 00:49:36.696 --> 00:49:40.176 A:middle Those two add to be negative 13x to the 4th. 00:49:41.486 --> 00:49:45.166 A:middle And then last times last we get a negative 12. 00:49:46.516 --> 00:49:49.076 A:middle So we have a correct factorization. 00:49:49.856 --> 00:49:54.546 A:middle However, we're not finished factoring with this one because we noticed 00:49:54.546 --> 00:49:58.566 A:middle that this first factor is the difference of two squares. 00:49:59.896 --> 00:50:04.706 A:middle We know that when we have a squared minus b squared, 00:50:05.526 --> 00:50:11.226 A:middle that's going to factor into a plus b times a minus b. 00:50:12.236 --> 00:50:20.296 A:middle Now if a squared is x to the 1/4, then that means a will be x to the 1/8. 00:50:20.936 --> 00:50:34.356 A:middle So this first factor will factor into x to the 1/8 plus 2 times x to the 1/8 minus 2. 00:50:35.856 --> 00:50:46.486 A:middle The second factor cannot be factored anymore so we just bring it down 4x to the 1/4 plus 3. 00:50:47.076 --> 00:50:51.586 A:middle And now, we have a complete factorization. 00:51:00.636 --> 00:51:06.026 A:middle In this example, we have to factor 6 times the quantity x plus y 00:51:06.026 --> 00:51:12.856 A:middle to the 6th plus 29 times the quantity x plus y to the 3rd plus 35. 00:51:13.456 --> 00:51:19.966 A:middle We verify that it's a quadratic type expression because the exponent 00:51:19.966 --> 00:51:23.776 A:middle of 6 is double the exponent of 3. 00:51:24.296 --> 00:51:34.266 A:middle We will do that temporary substitution where we let a equal x plus y to the 3rd, 00:51:37.316 --> 00:51:47.616 A:middle then a squared would be quantity x plus y to the 3rd that quantity squared 00:51:48.916 --> 00:51:53.356 A:middle which is quantity x plus y to the 6th. 00:51:54.976 --> 00:51:58.426 A:middle Now let's go ahead and do these substitutions 00:51:58.476 --> 00:52:10.046 A:middle and the expression becomes 6 times a squared plus 29 times a plus 35. 00:52:10.616 --> 00:52:16.066 A:middle We are electing to use trial and error in this objective. 00:52:17.336 --> 00:52:25.256 A:middle So we take a look at the factors of 6 and we know we could use 1 times 6 or 2 times 3. 00:52:26.656 --> 00:52:36.996 A:middle Next we look at the factors of 35 and that could be 1 times 35 or it could be 5 times 7. 00:52:39.926 --> 00:52:45.586 A:middle Very often we'll start out by trying the factors that are closest together. 00:52:46.326 --> 00:52:53.456 A:middle So let's do that and we would have 2a times 3a. 00:52:54.776 --> 00:53:00.606 A:middle For 35, we could use 5 times 7. 00:53:00.846 --> 00:53:04.386 A:middle And now it's time to check the cross products. 00:53:05.886 --> 00:53:08.916 A:middle We know that all the terms are going to be positive 00:53:09.046 --> 00:53:13.466 A:middle because the constant term is positive and the middle term is positive. 00:53:14.896 --> 00:53:19.806 A:middle So if we look at this first cross product, we get 15a. 00:53:20.726 --> 00:53:25.316 A:middle If we look at the second cross product, we get 14a. 00:53:26.046 --> 00:53:33.976 A:middle And when we add those two together, we have 29a and that matches our middle term. 00:53:34.806 --> 00:53:50.626 A:middle Now this means we can do the factorization as 2a plus 5 times 3a plus 7. 00:53:54.216 --> 00:54:02.616 A:middle Now remember, this was a temporary substitution to make things easier to see. 00:54:03.336 --> 00:54:09.336 A:middle And now, we have to return our variables to those in the original expression. 00:54:11.796 --> 00:54:21.216 A:middle So everywhere we see an a, we will put instead quantity x plus y to the 3rd power. 00:54:21.756 --> 00:54:33.616 A:middle So this first factor is going to become 2 times the quantity x plus y to the 3rd power plus 5. 00:54:34.156 --> 00:54:46.676 A:middle And the second factor is going to become 3 times the quantity x plus y to the 3rd power plus 7. 00:54:50.316 --> 00:54:53.836 A:middle Now we're ready to FOIL to check our answer. 00:54:55.116 --> 00:55:03.696 A:middle First time first gives us 6 times the quantity x plus y to the 6th. 00:55:03.856 --> 00:55:06.696 A:middle Next we do the outer and inner terms. 00:55:07.756 --> 00:55:17.146 A:middle We have 14 times the quantity x plus y cubed plus 15 times the quantity x plus y cubed 00:55:17.516 --> 00:55:23.386 A:middle which combine to give us the 29 times the quantity x plus y cubed. 00:55:23.866 --> 00:55:29.876 A:middle And then last times last we get 35. 00:55:30.386 --> 00:55:34.496 A:middle So this is a correct factorization. 00:55:43.776 --> 00:55:52.236 A:middle In this example, we will factor 15x to the 14th plus 44x to the 7th minus 20. 00:55:52.716 --> 00:56:00.606 A:middle We verify that this is a quadratic type expression because the highest exponent 00:56:00.606 --> 00:56:05.146 A:middle of 14 is double, the next highest exponent which is 7. 00:56:07.756 --> 00:56:16.506 A:middle We will use the technique where we do the temporary substitution and let a equal x 00:56:16.506 --> 00:56:27.336 A:middle to the 7th, then a squared would be x to the 7th squared which is x to the 14th. 00:56:29.076 --> 00:56:34.566 A:middle Now when we do the substitutions, this expression turns 00:56:34.566 --> 00:56:42.496 A:middle into 15a squared plus 44a minus 20. 00:56:43.056 --> 00:56:48.766 A:middle We are electing to use the trial and error method. 00:56:49.706 --> 00:56:58.666 A:middle So what we need consider, the factors of 15 which are 1 times 15 or 3 times 5, 00:56:59.746 --> 00:57:08.206 A:middle and the factors of 20 which are 1 times 20, 2 times 10, or 4 times 5. 00:57:10.706 --> 00:57:16.816 A:middle Very often we elect to start out by using the factors that are closer together. 00:57:17.846 --> 00:57:28.366 A:middle So let's try 3a times 5a and 4 times 5. 00:57:33.076 --> 00:57:35.686 A:middle Because our constant term is negative, 00:57:36.166 --> 00:57:40.676 A:middle we know that one of the cross products will negative and one will be positive. 00:57:41.726 --> 00:57:47.286 A:middle Because the middle term is positive, we know that we will make sure that the largest 00:57:47.736 --> 00:57:52.756 A:middle of the two cross products is the one that's positive and the smaller one is negative, 00:57:52.756 --> 00:57:56.846 A:middle so that when we add them together we end up with a positive number. 00:57:59.076 --> 00:58:08.766 A:middle The first cross product here we get 20 times a, the second cross product here is 15a. 00:58:09.636 --> 00:58:14.406 A:middle We would make the smaller one negative which makes this 5 negative. 00:58:14.946 --> 00:58:22.326 A:middle Now when we add these up, we get 5a which does not match our 44a 00:58:22.326 --> 00:58:25.206 A:middle that we need, so this one didn't work. 00:58:25.816 --> 00:58:31.736 A:middle We could try switching the 4 and the 5. 00:58:32.236 --> 00:58:42.846 A:middle So maybe we will keep the 3a and 5a the same and we'll put the 5 here and the 4 here. 00:58:43.976 --> 00:58:48.096 A:middle The first cross product is 25a. 00:58:48.706 --> 00:58:52.256 A:middle The second cross product is 12a. 00:58:53.176 --> 00:58:55.216 A:middle Make the smaller one negative. 00:58:56.216 --> 00:59:04.036 A:middle Let's add those up and we get 13a which still is not the 44a. 00:59:07.036 --> 00:59:10.076 A:middle Maybe we'll do 2 times 10 to get the 20. 00:59:11.336 --> 00:59:21.736 A:middle So we'll leave the 3a and the 5a alone and we'll try 2 times 10. 00:59:22.436 --> 00:59:28.706 A:middle These cross products give us 10a and 30a. 00:59:30.086 --> 00:59:31.846 A:middle Make the smaller one negative. 00:59:31.916 --> 00:59:40.726 A:middle Add those two up and we get 20a, which does not match the 44a. 00:59:41.286 --> 00:59:55.526 A:middle Let's try switching these two around, 3a times 5a, put the 10 here and the 2 here. 00:59:56.696 --> 00:59:59.976 A:middle This first cross product is 50a. 01:00:01.016 --> 01:00:04.446 A:middle The second cross product is 6a. 01:00:05.556 --> 01:00:07.806 A:middle Make the smaller one be negative. 01:00:09.596 --> 01:00:17.746 A:middle When we add these two up, we get 44a which is exactly the middle term that we needed. 01:00:18.256 --> 01:00:23.606 A:middle So that tells us that we can create our factors 01:00:24.206 --> 01:00:39.686 A:middle to be quantity 3a plus 10 times the quantity 5a minus 2. 01:00:43.796 --> 01:00:48.246 A:middle Now this was a temporary substitution that we did. 01:00:50.116 --> 01:00:54.636 A:middle We will need to return to the variables of the original expression, 01:00:55.156 --> 01:01:02.016 A:middle so everywhere that we see an a we will be replacing that with an x to the 7th. 01:01:02.436 --> 01:01:12.366 A:middle So this becomes in the first factor, we will have 3 times x to the 7th plus 10. 01:01:12.926 --> 01:01:20.466 A:middle And then the second factor, we will have 5 times x to the 7th minus 2. 01:01:23.156 --> 01:01:25.626 A:middle Let's FOIL to check our answer. 01:01:26.706 --> 01:01:31.976 A:middle First times first gives us 15x to the 14th. 01:01:33.256 --> 01:01:36.656 A:middle The outer and inner terms will combine. 01:01:37.166 --> 01:01:46.856 A:middle We have negative 6x to the 7th plus 50x to the 7th which adds to 44x to the 7th. 01:01:47.656 --> 01:01:51.146 A:middle Then last times last is negative 20. 01:01:52.386 --> 01:01:54.906 A:middle So we have a correct factorization. 01:02:05.116 --> 01:02:07.406 A:middle It's time to check your understanding. 01:02:07.946 --> 01:02:12.266 A:middle So pause the video to try this one on your own, then restart the video 01:02:12.266 --> 01:02:14.506 A:middle when you are ready to check your answer. 01:02:18.976 --> 01:02:28.406 A:middle In this example we have to factor 12x to the 4/3 minus 28x to the 2/3 plus 15. 01:02:28.936 --> 01:02:35.356 A:middle We verify that this is a quadratic type expression because the exponent 01:02:35.356 --> 01:02:39.586 A:middle of 4/3 is double the exponent of 2/3. 01:02:41.456 --> 01:02:49.676 A:middle We will do a temporary substitution where we let a equal x to the 2/3, 01:02:52.276 --> 01:03:00.156 A:middle then a squared would be x to the 2/3 squared. 01:03:01.306 --> 01:03:04.856 A:middle Remember, when you take a power to a power, you multiply. 01:03:05.236 --> 01:03:08.366 A:middle So this becomes x to the 4/3. 01:03:08.976 --> 01:03:13.516 A:middle And now we can do these substitutions. 01:03:14.846 --> 01:03:24.816 A:middle The original expression becomes 12a squared minus 28a plus 15. 01:03:26.406 --> 01:03:35.336 A:middle Using trial and error, we need the factors of 12 which are 1 times 12, 2 times 6, 01:03:36.386 --> 01:03:46.506 A:middle and 3 times 4 as well as the factors of 15 which are 1 times 15 or 3 times 5. 01:03:47.096 --> 01:03:52.576 A:middle It's common to start off using the factors that are closer together. 01:03:53.066 --> 01:04:03.696 A:middle So we can try 3a times 4a and 3 times 5 for the constant term. 01:04:07.436 --> 01:04:12.776 A:middle Now because our constant term is positive but our middle term is negative, 01:04:13.416 --> 01:04:17.016 A:middle we know that both factors will have to be negative. 01:04:17.486 --> 01:04:21.196 A:middle And let's check the cross products. 01:04:21.666 --> 01:04:29.676 A:middle This first one gives us negative 12a, the second one gives us negative 15a, 01:04:30.456 --> 01:04:38.716 A:middle and those add to be negative 27a but that doesn't match our middle term of negative 28a. 01:04:41.176 --> 01:04:44.176 A:middle Maybe we would try switching these two. 01:04:45.436 --> 01:04:54.976 A:middle So we have 3a and 4a again but we'll try switching the negative 5 and the negative 3. 01:04:57.016 --> 01:05:06.586 A:middle The first cross product is negative 20a, the second cross product is negative 9a which add 01:05:06.586 --> 01:05:12.546 A:middle to be negative 29a, again, not quite what we need. 01:05:14.256 --> 01:05:16.666 A:middle Maybe we'll try switching up. 01:05:16.986 --> 01:05:23.046 A:middle Instead of 3a and 4a, maybe we'll try 2a and 6a. 01:05:24.746 --> 01:05:33.056 A:middle So if we do those and we'll try the negative 3 and the negative 5 here, 01:05:34.156 --> 01:05:43.026 A:middle the first cross product is negative 18a, the second cross product is negative 10a, 01:05:43.766 --> 01:05:50.386 A:middle and those two add to be negative 28a which is exactly what we need. 01:05:50.896 --> 01:05:56.716 A:middle So that tells us we can do our factorization here. 01:06:00.496 --> 01:06:15.256 A:middle The first factor is going to be 2a minus 3 and the second factor is going to be 6a minus 5. 01:06:18.896 --> 01:06:26.856 A:middle Now again, this was a temporary substitution and we need to return 01:06:26.856 --> 01:06:29.426 A:middle to the variables in the original expression. 01:06:30.996 --> 01:06:36.646 A:middle Everywhere we see an a, we will replace it with an x to the 2/3. 01:06:37.256 --> 01:06:51.846 A:middle So this becomes 2x to the 2/3 minus 3 times 6x to the 2/3 minus 5. 01:06:52.326 --> 01:06:58.026 A:middle Now we are ready to FOIL to check our answer. 01:06:58.716 --> 01:07:03.036 A:middle First times first gives us 12x to the 4/3. 01:07:04.346 --> 01:07:12.936 A:middle Combining the outer and inner terms, we have a negative 10x to the 2/3 plus a negative 18x 01:07:12.936 --> 01:07:17.396 A:middle to the 2/3, which gives us our negative 28x to the 2/3. 01:07:18.716 --> 01:07:22.896 A:middle And then last times last we get a positive 15. 01:07:24.136 --> 01:07:26.736 A:middle So this is a correct factorization. 01:07:32.606 --> 01:07:36.186 A:middle Below are some practice problems you can try on your own. 01:07:37.046 --> 01:07:42.906 A:middle You may either pause the video to work on them now or write them down to work on them later. 01:07:43.986 --> 01:07:47.386 A:middle After a few seconds, the answers will be revealed. 01:07:55.636 --> 01:07:57.986 A:middle In objective 4, you will learn how 01:07:57.986 --> 01:08:02.666 A:middle to factor quadratic type expressions with the greatest common factor. 01:08:03.236 --> 01:08:12.446 A:middle In this example, we have to factor 24x to the 7th y squared minus 20x 01:08:12.446 --> 01:08:18.076 A:middle to the 4th y to the 4th plus 4xy to the 6th. 01:08:18.476 --> 01:08:31.866 A:middle Looking at these three terms, we see that the greatest common factor is 4xy squared. 01:08:32.406 --> 01:08:43.356 A:middle So we will factor that out from all three terms, 4xy squared. 01:08:44.246 --> 01:08:52.636 A:middle And what we have left, just do the division, 24x to the 7th y squared divided 01:08:52.636 --> 01:09:04.316 A:middle by 4xy squared is 6x to the 6th minus, if we divide 20x to the 4th y to the 4th 01:09:04.386 --> 01:09:12.286 A:middle by 4xy squared, we get 5x cubed y squared. 01:09:12.886 --> 01:09:25.766 A:middle And if we divide 4xy to the 6th by 4xy squared, we get y to the 4th. 01:09:26.996 --> 01:09:36.146 A:middle Now in this trinomial, we notice that we have a quadratic type expression because the exponent 01:09:36.146 --> 01:09:42.266 A:middle of 6 is double the exponent of 3 and the exponent of 4 01:09:42.486 --> 01:09:47.646 A:middle and the y variable doubles the exponent of 2 in the y variable. 01:09:48.576 --> 01:09:57.376 A:middle So we will do that temporary substitution where we let a equal x cubed, 01:09:58.496 --> 01:10:08.086 A:middle which means a squared would be x cubed squared which is x to the 6th. 01:10:10.576 --> 01:10:17.836 A:middle We also need a second substitution where we let b equal y squared, 01:10:19.146 --> 01:10:27.986 A:middle then b squared will be y squared, squared which is y to the 4th. 01:10:29.136 --> 01:10:34.966 A:middle Next we will substitute these temporary variables into the original expression. 01:10:36.126 --> 01:10:40.466 A:middle So let's bring down that 4xy squared. 01:10:40.466 --> 01:10:42.096 A:middle We're just going to leave that alone. 01:10:43.256 --> 01:10:48.276 A:middle We're going to do the temporary substitution here on this trinomial. 01:10:48.766 --> 01:10:59.176 A:middle So this becomes 6a squared minus 5ab plus b squared. 01:10:59.646 --> 01:11:05.986 A:middle Let's go ahead and practice trial and error method. 01:11:06.706 --> 01:11:14.496 A:middle So what we'll need to do is come up with the factors of 6 which are either 1 times 6 01:11:14.896 --> 01:11:25.226 A:middle or 2 times 3, and the factors of 1 here which will only be 1 times 1. 01:11:25.446 --> 01:11:30.616 A:middle So, it's often customary to try the two factors that are closest together. 01:11:31.206 --> 01:11:35.586 A:middle So let's try 2a times 3a. 01:11:36.196 --> 01:11:40.476 A:middle And then of course the only way to get b squared is going 01:11:40.476 --> 01:11:50.656 A:middle to be b times b. Now the third term is positive, which tells us that these two will have 01:11:50.656 --> 01:11:55.666 A:middle to have the same sign, because the middle term is negative we know 01:11:55.666 --> 01:11:58.496 A:middle that both of these will be negative. 01:11:59.756 --> 01:12:08.056 A:middle And now let's check the cross products, 3a times a negative b is negative 3ab, 01:12:09.036 --> 01:12:15.006 A:middle and the second cross product, 2a times negative b is negative 2ab. 01:12:16.086 --> 01:12:24.206 A:middle When we add those up, we get negative 5ab which is exactly what we needed in the middle term. 01:12:25.996 --> 01:12:32.896 A:middle So let's continue to carry down the 4xy squared. 01:12:33.436 --> 01:12:39.606 A:middle And now we have our factorization of this already. 01:12:40.176 --> 01:12:50.906 A:middle The first factor is going to be 2a minus b and the second factor is going 01:12:50.906 --> 01:13:01.266 A:middle to be 3a minus b. Now remember, this was a temporary substitution that we did. 01:13:01.866 --> 01:13:08.196 A:middle So we will need to return to the variables from the original expression 01:13:10.136 --> 01:13:16.106 A:middle that 4xy squared is already in the original variables. 01:13:17.376 --> 01:13:23.566 A:middle But now, everywhere that we see an a, we will be replacing that with x cubed 01:13:24.236 --> 01:13:29.306 A:middle and everywhere we see a b, we will be replacing that with y squared. 01:13:29.846 --> 01:13:44.686 A:middle So the factorization becomes 2 times x cubed minus y squared times the quantity 3 times x 01:13:44.686 --> 01:13:47.586 A:middle cubed minus y squared. 01:13:50.936 --> 01:13:55.296 A:middle Now we are ready to FOIL to check our answer. 01:13:55.366 --> 01:14:01.086 A:middle And we'll be checking with this trinomial up here. 01:14:02.156 --> 01:14:08.696 A:middle First times first gives us 6x to the 6th. 01:14:08.696 --> 01:14:19.286 A:middle The outer and inner terms combine to be negative 2x cubed y squared minus 3x cubed y squared 01:14:19.556 --> 01:14:23.746 A:middle which is the negative 5x cubed y squared. 01:14:24.766 --> 01:14:29.706 A:middle And then last times last gives us a plus y to the 4th. 01:14:31.056 --> 01:14:34.036 A:middle So we have a correct factorization. 01:14:43.816 --> 01:14:48.816 A:middle In this example, we have to factor 30y times the quantity x minus 4 01:14:48.816 --> 01:14:57.136 A:middle to the 6th plus 27y times the quantity x minus 4 to the 3rd minus 27y. 01:14:57.716 --> 01:15:04.216 A:middle We can see that these three terms do have a greatest common factor, 01:15:06.356 --> 01:15:12.926 A:middle which is equal to 3 times y. So we're going to factor that out right from the beginning, 01:15:14.436 --> 01:15:18.676 A:middle 3y and then we will have the quantity. 01:15:19.416 --> 01:15:21.086 A:middle So let's do the division here. 01:15:21.376 --> 01:15:29.566 A:middle This first term divided by 3y becomes 10 times the quantity x minus 4 01:15:29.956 --> 01:15:42.546 A:middle to the 6th plus the second term divided by 3y becomes 9 times the quantity x minus 4 01:15:42.876 --> 01:15:51.456 A:middle to the 3rd minus the third term divided by 3y becomes 9. 01:15:52.066 --> 01:15:59.516 A:middle Now we recognize this is a quadratic type expression because the exponent 01:15:59.516 --> 01:16:03.066 A:middle of 6 is double the exponent of 3. 01:16:04.636 --> 01:16:16.066 A:middle So we'll use the technique where we let a equal the quantity x minus 4 to the 3rd power, 01:16:17.506 --> 01:16:25.966 A:middle then a squared would be quantity x minus 4 01:16:26.506 --> 01:16:34.516 A:middle to the 3rd power squared which is x minus 4 to the 6th. 01:16:34.996 --> 01:16:41.136 A:middle And now, we can do those substitutions into this trinomial. 01:16:43.596 --> 01:16:59.136 A:middle So first, we bring down the 3y, and then the trinomial becomes 10a squared plus 9a minus 9. 01:16:59.686 --> 01:17:09.866 A:middle And once again, let's practice using trial and error, so we will need the factors of 10 01:17:10.436 --> 01:17:14.786 A:middle which are 1 times 10 or 2 times 5. 01:17:15.866 --> 01:17:22.446 A:middle We need the factors of 9 which are 1 times 9 or 3 times 3. 01:17:22.946 --> 01:17:28.456 A:middle Let's start off using the two factors that are closer together. 01:17:29.256 --> 01:17:33.436 A:middle And we have 2a times 5a. 01:17:34.016 --> 01:17:40.486 A:middle We'll try these two, 3 times 3 because those are closer together. 01:17:41.086 --> 01:17:44.906 A:middle Now because the constant term is negative, 01:17:45.436 --> 01:17:49.396 A:middle we know that one of these will be negative and one will be positive. 01:17:51.116 --> 01:17:53.296 A:middle The middle term is positive. 01:17:54.066 --> 01:17:56.366 A:middle So when we check our cross products, 01:17:56.706 --> 01:18:01.156 A:middle we will want to make the larger magnitude cross product be the positive 01:18:01.516 --> 01:18:03.316 A:middle and the smaller one be the negative. 01:18:04.736 --> 01:18:13.986 A:middle This first cross product is 15a, and the second cross product is 6a. 01:18:14.996 --> 01:18:20.156 A:middle Now remember, we want the larger one to be positive and we'll take the smaller one 01:18:20.676 --> 01:18:27.586 A:middle and change it to a negative, which means we have to make that 3 negative. 01:18:28.836 --> 01:18:33.636 A:middle Now let's add these two up and we get 9a 01:18:34.646 --> 01:18:38.436 A:middle which is exactly what we needed for our middle term over here. 01:18:39.436 --> 01:18:46.406 A:middle So now we know we have the factors, let's go ahead and carry this down here. 01:18:47.356 --> 01:18:48.756 A:middle We've got the 3y. 01:18:49.376 --> 01:19:10.496 A:middle And now, this trinomial will factor as 2a plus 3 times the quantity 5a minus 3. 01:19:13.756 --> 01:19:18.726 A:middle Remember that this all was a temporary substitution. 01:19:19.306 --> 01:19:25.096 A:middle We will need to return to the variables of the original expression. 01:19:25.686 --> 01:19:32.396 A:middle So this becomes 3y which is already in the original variables. 01:19:33.496 --> 01:19:37.856 A:middle Everywhere we see the a variable, we are going to trade it 01:19:38.346 --> 01:19:41.476 A:middle for the quantity x minus 4 to the 3rd power. 01:19:42.706 --> 01:19:48.766 A:middle So this become 2 times the quantity x minus 4 01:19:49.126 --> 01:20:03.586 A:middle to the 3rd plus 3 times 5 times the quantity x minus 4 to the 3rd minus 3. 01:20:06.526 --> 01:20:08.876 A:middle Now let's FOIL to check our answer. 01:20:10.406 --> 01:20:18.276 A:middle First times first gives us 10 times the quantity x minus 4 to the 6th which we see here. 01:20:19.656 --> 01:20:22.346 A:middle Combining the outer and inner terms, 01:20:22.656 --> 01:20:28.786 A:middle we get a minus 6 times the quantity plus 15 times the quantity, 01:20:28.926 --> 01:20:33.786 A:middle which gives us 9 times the quantity x minus 4 to the 3rd. 01:20:34.536 --> 01:20:38.676 A:middle And then last times last gives us that minus 9. 01:20:39.626 --> 01:20:41.976 A:middle So we have a correct factorization. 01:20:50.836 --> 01:21:00.736 A:middle In this example, we have to factor 20x to the 3/2 minus 31x to the 5/4 minus 9x. 01:21:01.276 --> 01:21:07.916 A:middle We notice right away that these three terms have a greatest common factor 01:21:07.916 --> 01:21:12.986 A:middle of x. So let's factor that out. 01:21:17.756 --> 01:21:28.106 A:middle If we take 20x to the 3/2 and divide it by x, we get 20x to the 1/2 minus, 01:21:29.316 --> 01:21:36.906 A:middle if we divide 31x to the 5/4 by x, we're going to get 31x 01:21:37.636 --> 01:21:46.406 A:middle to the 1/4 minus, and then 9x divided by x is 9. 01:21:50.456 --> 01:21:56.976 A:middle We notice right away that this is a quadratic type expression because the exponent 01:21:56.976 --> 01:22:00.776 A:middle of 1/2 is double the exponent of 1/4. 01:22:01.896 --> 01:22:09.216 A:middle So let's do that temporary substitution where we let a equal x to the 1/4, 01:22:10.456 --> 01:22:25.566 A:middle then a squared will be x to the 1/4 squared which is x to the 2/4 which is x to the 1/2. 01:22:25.766 --> 01:22:33.706 A:middle We continue to bring down this x. And if we do the substitutions, 01:22:35.046 --> 01:22:43.936 A:middle our trinomial becomes 20a squared minus 31a minus 9. 01:22:44.506 --> 01:22:52.186 A:middle Let's go ahead and practice factoring with trial and error. 01:22:53.286 --> 01:23:02.256 A:middle The factors of 20 would be 1 times 20, 2 times 10, or 4 times 5. 01:23:02.856 --> 01:23:10.366 A:middle The factors of 9 would be 1 times 9 or 3 times 3. 01:23:10.886 --> 01:23:15.836 A:middle Let's start with the two factors that are closest together. 01:23:16.836 --> 01:23:21.276 A:middle And try 4a times 5a. 01:23:21.876 --> 01:23:28.196 A:middle And if we start with 3 times 3 to make the 9. 01:23:29.546 --> 01:23:35.506 A:middle Now we have to be careful about our positive and negative signs because we need a negative 01:23:35.996 --> 01:23:41.026 A:middle for the constant term, which means one of these will be negative and one will be positive. 01:23:41.966 --> 01:23:44.416 A:middle We need a negative for the middle terms. 01:23:45.136 --> 01:23:50.326 A:middle So that tells us that when we do our cross products, we'll want to make the larger 01:23:50.326 --> 01:23:53.286 A:middle of the two cross products the one that's negative. 01:23:54.236 --> 01:23:56.036 A:middle So let's see what we have here. 01:23:56.556 --> 01:24:04.346 A:middle The first cross product becomes 15a, the second one is 12a. 01:24:05.116 --> 01:24:10.706 A:middle If we make the larger one negative, then that means we have to make this 3 negative. 01:24:11.246 --> 01:24:17.466 A:middle Now when we add these two together, we get negative 3a 01:24:17.956 --> 01:24:23.126 A:middle which definitely does not match the negative 31a that we need. 01:24:24.546 --> 01:24:29.216 A:middle And we can see even if we switched the signs of the 3s, 01:24:29.856 --> 01:24:34.216 A:middle then all that would do is make this a positive 3a. 01:24:34.726 --> 01:24:36.236 A:middle So that's not going to work. 01:24:36.956 --> 01:24:42.126 A:middle So we either have to change the 4 and the 5 or we have to change the 3s. 01:24:43.056 --> 01:24:48.876 A:middle You don't really know which one is going to be, but maybe we'll try leaving the 4 and 5 the same 01:24:49.596 --> 01:24:52.996 A:middle but put a 1 and a 9 for the constant term. 01:24:54.046 --> 01:25:02.796 A:middle So we've got 4a times 5a, and we will put a 1 and 9. 01:25:04.296 --> 01:25:06.616 A:middle Now let's check the cross products. 01:25:07.266 --> 01:25:10.406 A:middle This first one would be 5a. 01:25:11.156 --> 01:25:15.196 A:middle The second one would be 36a. 01:25:15.876 --> 01:25:20.676 A:middle We want the larger to be negative which mean the 9 has to be negative. 01:25:21.576 --> 01:25:30.636 A:middle Now let's check our addition here, 5a minus 36a is negative 31a 01:25:31.346 --> 01:25:35.526 A:middle which is exactly what we needed here for the middle term. 01:25:36.086 --> 01:25:45.426 A:middle So if we continue here, this x comes straight down, and now we can factor this trinomial 01:25:46.196 --> 01:25:59.736 A:middle into these two factors, we've got 4a plus 1 times 5a minus 9. 01:26:03.536 --> 01:26:12.216 A:middle Now remember, this was a temporary substitution that we did, so we will need to return 01:26:12.216 --> 01:26:20.796 A:middle to the variables from the original expression and this becomes x. Now everywhere we see an a 01:26:20.796 --> 01:26:27.016 A:middle in these two factors, we will have to replace it with x to the 1/4. 01:26:28.246 --> 01:26:35.146 A:middle So this first factor becomes 4x to the 1/4 plus 1. 01:26:36.556 --> 01:26:43.716 A:middle The second factor becomes 5x to the 1/4 minus 9. 01:26:44.286 --> 01:26:48.756 A:middle And now let's FOIL to check our answer. 01:26:49.636 --> 01:26:55.986 A:middle First times first is 20x to the 1/2. 01:26:56.196 --> 01:27:05.376 A:middle Combining the outer term and the inner term, we get negative 36x to the 4th plus 5x to the 4th, 01:27:06.336 --> 01:27:09.006 A:middle which is negative 31x to the 4th. 01:27:09.436 --> 01:27:13.996 A:middle And then last times last is negative 9. 01:27:14.586 --> 01:27:17.976 A:middle So we have a correct factorization. 01:27:27.736 --> 01:27:30.136 A:middle It's time to check your understanding. 01:27:30.476 --> 01:27:35.006 A:middle So pause the video to try this one on your own, then restart the video 01:27:35.006 --> 01:27:36.926 A:middle when you are ready to check your answer. 01:27:42.376 --> 01:27:50.706 A:middle In this example, we have to factor 12x to the 9th minus 46x to the 5th plus 30x. 01:27:51.276 --> 01:27:59.736 A:middle We see that these three terms have a greatest common factor of 2x. 01:28:03.136 --> 01:28:06.066 A:middle So let's factor that out first. 01:28:06.676 --> 01:28:13.156 A:middle If we take the first term and divide it by 2x, 01:28:13.636 --> 01:28:24.306 A:middle we will have 6x to the 8th minus the second term divided by 2x is going to be 23x 01:28:24.786 --> 01:28:33.036 A:middle to the 4th plus the third term divided 2x will be 15. 01:28:35.876 --> 01:28:41.076 A:middle Now we see that this is a quadratic type expression because the exponent 01:28:41.076 --> 01:28:44.616 A:middle of 8 is double the exponent of 4. 01:28:46.026 --> 01:28:53.176 A:middle So let's do that temporary substitution where we let a equal x to the 4th, 01:28:54.256 --> 01:29:04.266 A:middle then a squared would be x to the 4th squared which is x to the 8th. 01:29:04.516 --> 01:29:09.086 A:middle Here we're going to bring down that factor 2x. 01:29:09.616 --> 01:29:21.506 A:middle And then in the trinomial, we will substitute and get 6a squared minus 23a plus 15. 01:29:22.066 --> 01:29:27.376 A:middle Let's go ahead and factor by trial and error. 01:29:28.656 --> 01:29:34.686 A:middle The factors of 6 are 1 times 6 or 2 times 3. 01:29:35.466 --> 01:29:42.206 A:middle The factors of 15 are 1 times 15 or 3 times 5. 01:29:42.786 --> 01:29:48.426 A:middle As this customary, we'll start off using the factors closest together, 01:29:49.166 --> 01:29:52.806 A:middle so we have 2a times 3a. 01:29:53.376 --> 01:30:00.516 A:middle And for the 15, we'll use 3 times 5. 01:30:02.666 --> 01:30:05.616 A:middle Now let's think about the positive and negative signs. 01:30:06.186 --> 01:30:08.336 A:middle We see the constant term is positive, 01:30:09.076 --> 01:30:12.976 A:middle which means our two factors have to have the same sign. 01:30:13.606 --> 01:30:18.586 A:middle The middle term is negative, so that means they are both going to be negative. 01:30:19.626 --> 01:30:21.506 A:middle And let's check the cross products. 01:30:22.266 --> 01:30:30.356 A:middle The first cross product is a negative 9a, the second cross product is a negative 10a, 01:30:31.186 --> 01:30:41.366 A:middle and those add up to be negative 19a which does not match the negative 23a that we need. 01:30:42.216 --> 01:30:45.276 A:middle Well, maybe we'll try switching these two around. 01:30:45.986 --> 01:30:57.336 A:middle So keep the 2a and 3a but switch the negative 5 with the negative 3. 01:30:58.226 --> 01:31:06.546 A:middle The first cross product is negative 15a, the second cross product is negative 6a, 01:31:07.566 --> 01:31:16.646 A:middle and those two add to be negative 21a which also does not match the negative 23a. 01:31:17.196 --> 01:31:26.816 A:middle So maybe we will try 1 times 6, so we have a times 6a. 01:31:27.446 --> 01:31:32.936 A:middle And we'll try the negative 3 times the negative 5. 01:31:34.256 --> 01:31:43.676 A:middle The first cross product is negative 18a, the second cross product is negative 5a, 01:31:45.046 --> 01:31:53.666 A:middle and those two add to be negative 23a which matches what we need here in the middle term. 01:31:54.246 --> 01:31:58.676 A:middle So let's continue. 01:31:58.676 --> 01:32:17.156 A:middle We bring down the 2x and then we factor this trinomial into a minus 3 times 6a minus 5. 01:32:21.716 --> 01:32:30.246 A:middle Remember that this was a temporary substitution, so we will need to return 01:32:30.246 --> 01:32:32.816 A:middle to the variables of the original expression. 01:32:33.356 --> 01:32:40.266 A:middle We bring down that 2x which is already in the original variables. 01:32:41.526 --> 01:32:47.516 A:middle Everywhere we see an a, we are going to replace it with x to the 4th, 01:32:48.496 --> 01:33:04.086 A:middle so this factorization becomes x to the 4th minus 3 times the quantity 6x to the 4th minus 5. 01:33:04.296 --> 01:33:06.636 A:middle Let's FOIL to check our answer. 01:33:07.606 --> 01:33:11.096 A:middle First times first is 6x to the 8th. 01:33:12.326 --> 01:33:20.256 A:middle Combining the outer and inner terms, we get negative 5x to the 4th minus 18x to the 4th 01:33:20.376 --> 01:33:23.416 A:middle which is negative 23x to the 4th. 01:33:24.306 --> 01:33:31.666 A:middle And then last times last is a positive 15, so this is a correct factorization. 01:33:37.306 --> 01:33:40.926 A:middle Below are some practice problems you can try on your own. 01:33:41.786 --> 01:33:47.636 A:middle You may either pause the video to work on them now or write them down to work on them later. 01:33:48.716 --> 01:33:51.976 A:middle After a few seconds, the answers will be revealed.