1 00:00:00,000 --> 00:00:00,760 2 00:00:00,760 --> 00:00:04,519 I've been asked to make a video on algebraic division or 3 00:00:04,519 --> 00:00:05,769 algebraic long division. 4 00:00:05,769 --> 00:00:09,779 So I'll make a video an algebraic long division. 5 00:00:09,779 --> 00:00:10,929 I'm just going to make up a problem. 6 00:00:10,929 --> 00:00:14,759 Let's say we wanted to divide-- we wanted to see how many times 7 00:00:14,759 --> 00:00:17,699 does-- I'll start with a fairly straightforward problem. 8 00:00:17,699 --> 00:00:27,679 How many times does 2x plus 1 go into-- I don't know-- let's 9 00:00:27,679 --> 00:00:41,719 say it's 8x to the third minus 7x squared plus 10x minus 5. 10 00:00:41,719 --> 00:00:44,729 So what we do is we just take-- actually, just the exact same 11 00:00:44,729 --> 00:00:47,469 way that you would do with long division, traditional long 12 00:00:47,469 --> 00:00:50,390 division of multiple digits. 13 00:00:50,390 --> 00:00:52,640 In the 2x plus 1 expression you look at, oh, what is 14 00:00:52,640 --> 00:00:53,880 the highest degree term? 15 00:00:53,880 --> 00:00:55,859 And that's really all we're going to pay attention 16 00:00:55,859 --> 00:00:57,089 to most of the time. 17 00:00:57,090 --> 00:00:59,160 So the first step is you say, OK, the highest 18 00:00:59,159 --> 00:01:01,209 degree term is 2x. 19 00:01:01,210 --> 00:01:05,450 How many times does 2x go into the highest degree term of the 20 00:01:05,450 --> 00:01:07,850 number-- not the number-- the expression that we're 21 00:01:07,849 --> 00:01:09,009 dividing into? 22 00:01:09,010 --> 00:01:14,130 So you say, how many times does 2x go into 8x to the third? 23 00:01:14,129 --> 00:01:15,949 Well, we could do a little division on the side, but you 24 00:01:15,950 --> 00:01:18,320 could imagine eventually this is pretty straightforward. 25 00:01:18,319 --> 00:01:22,859 So if you have 8x to the third divided by 2x, that 26 00:01:22,859 --> 00:01:27,370 is equal to 4x squared. 27 00:01:27,370 --> 00:01:31,990 So 2x goes into 8x to the third 4x squared times. 28 00:01:31,989 --> 00:01:33,089 And this is the key thing. 29 00:01:33,090 --> 00:01:34,680 You don't want to write the 4x squared here. 30 00:01:34,680 --> 00:01:36,930 You want to keep everything in the correct places. 31 00:01:36,930 --> 00:01:39,730 So when you're dividing numbers you think of the ones, the 32 00:01:39,730 --> 00:01:43,160 tens, the hundreds, and the thousands place et cetera. 33 00:01:43,159 --> 00:01:44,859 When you're dividing polynomials you can kind of 34 00:01:44,859 --> 00:01:47,179 think of the x to the 0 space. 35 00:01:47,180 --> 00:01:49,860 The x to the 1 space or the x space. 36 00:01:49,859 --> 00:01:51,049 The x squared space. 37 00:01:51,049 --> 00:01:52,689 The x to the third space. 38 00:01:52,689 --> 00:01:58,560 So, when we say that 2x goes into 8x to the third 4x squared 39 00:01:58,560 --> 00:02:02,752 times let's write that in the x squared spot. 40 00:02:02,751 --> 00:02:08,409 It goes into it 4x squared times. 41 00:02:08,409 --> 00:02:10,905 Now, we take that 4x squared and we multiply 42 00:02:10,905 --> 00:02:11,990 it by our expression. 43 00:02:11,990 --> 00:02:14,740 I think you're already seeing that this is very similar 44 00:02:14,740 --> 00:02:16,180 to long division. 45 00:02:16,180 --> 00:02:19,469 And actually, if x was a ten, it would be identical 46 00:02:19,469 --> 00:02:20,530 to long division. 47 00:02:20,530 --> 00:02:22,460 And I'll let you think about that. 48 00:02:22,460 --> 00:02:24,360 If x was 10 this would be the thousands place. 49 00:02:24,360 --> 00:02:27,160 This would be 8,000 minus-- although you would have 50 00:02:27,159 --> 00:02:30,139 negative digits, which doesn't make a bunch of sense. 51 00:02:30,139 --> 00:02:31,539 But I think you get what I'm saying. 52 00:02:31,539 --> 00:02:35,729 But anyway, back to this algebraic long division. 53 00:02:35,729 --> 00:02:37,349 Although I think it is very important to see the 54 00:02:37,349 --> 00:02:40,629 parallels between this and traditional long division. 55 00:02:40,629 --> 00:02:42,599 Well, anyway, we said that 2x goes into 8x to the 56 00:02:42,599 --> 00:02:44,439 third 4x squared times. 57 00:02:44,439 --> 00:02:47,079 Now what we can do is we can multiply 4x squared 58 00:02:47,080 --> 00:02:49,040 times 2x plus 1. 59 00:02:49,039 --> 00:02:53,060 So 4x squared times 1, that's 4x squared. 60 00:02:53,060 --> 00:02:56,170 So we can write that in the x squared's place. 61 00:02:56,169 --> 00:02:59,369 We could write it 4x squared. 62 00:02:59,370 --> 00:03:04,349 And 4x squared times 2x is 8x to the third. 63 00:03:04,349 --> 00:03:06,924 64 00:03:06,925 --> 00:03:08,469 This is plus here. 65 00:03:08,469 --> 00:03:12,590 And now, just like we do with traditional long division, we 66 00:03:12,590 --> 00:03:15,509 can subtract this from this. 67 00:03:15,509 --> 00:03:26,754 So minus 7x squared minus 4x squared is minus 11x squared. 68 00:03:26,754 --> 00:03:29,990 69 00:03:29,990 --> 00:03:33,205 And then 8x to the third minus 8x to the third is 0, so we 70 00:03:33,205 --> 00:03:34,680 can ignore that right there. 71 00:03:34,680 --> 00:03:39,409 And if we want, we can bring down the rest of the number, 72 00:03:39,409 --> 00:03:41,990 but maybe just for fun we'll bring down the next spot just 73 00:03:41,990 --> 00:03:46,219 like we do in traditional long division. 74 00:03:46,219 --> 00:03:49,150 Actually, let me erase this over here. 75 00:03:49,150 --> 00:03:53,060 76 00:03:53,060 --> 00:03:57,120 Because I think we might find that real estate useful. 77 00:03:57,120 --> 00:03:59,730 All right, I'm back. 78 00:03:59,729 --> 00:04:01,969 Actually, it doesn't hurt to bring down the whole thing. 79 00:04:01,969 --> 00:04:05,539 Just so that you understand what we're doing. 80 00:04:05,539 --> 00:04:08,509 We're saying if you were to divide 2x plus 1 to this entire 81 00:04:08,509 --> 00:04:11,689 expression, and you say it goes in 4x squared times. 82 00:04:11,689 --> 00:04:14,539 Now you can kind of call it our intermediate 83 00:04:14,539 --> 00:04:16,909 remainder is what's left. 84 00:04:16,910 --> 00:04:17,800 This is what's left. 85 00:04:17,800 --> 00:04:20,949 You could almost imagine 4x squared times 2x plus 1 is-- 86 00:04:20,949 --> 00:04:26,189 this is 8x to the third plus 4x squared plus 0 plus 0 because 87 00:04:26,189 --> 00:04:28,600 it doesn't contribute any thing to these spots. 88 00:04:28,600 --> 00:04:30,740 But then what's left over is this expression. 89 00:04:30,740 --> 00:04:33,920 If you take this minus this whole expression, you 90 00:04:33,920 --> 00:04:35,810 get what's left over. 91 00:04:35,810 --> 00:04:37,790 Now we just do the same thing over. 92 00:04:37,790 --> 00:04:40,100 How many times does 2x-- we just look at the 93 00:04:40,100 --> 00:04:41,510 highest order term. 94 00:04:41,509 --> 00:04:47,180 How many times does 2x go into negative 11x squared? 95 00:04:47,180 --> 00:04:50,420 So let's write it here on the side again. 96 00:04:50,420 --> 00:04:52,009 Actually, let me do it here. 97 00:04:52,009 --> 00:04:58,930 So if we were to take minus 11x squared divided by 2x, 98 00:04:58,930 --> 00:04:59,780 that is equal to what? 99 00:04:59,779 --> 00:05:03,487 That is equal to minus 11/2 x. 100 00:05:03,487 --> 00:05:06,550 101 00:05:06,550 --> 00:05:13,389 So 2x goes into minus 11x squared minus 11/2 x times. 102 00:05:13,389 --> 00:05:15,509 So we'll write that in our x place. 103 00:05:15,509 --> 00:05:17,870 So minus 11/2. 104 00:05:17,870 --> 00:05:19,750 We could write that as 5.5. 105 00:05:19,750 --> 00:05:21,889 I'll just write it as a fraction. 106 00:05:21,889 --> 00:05:25,459 Minus 11/2 x. 107 00:05:25,459 --> 00:05:30,250 And now, what is minus 11/2 x times 2x plus 1? 108 00:05:30,250 --> 00:05:36,019 So minus 11/2 x times 1 is minus 11/2 x. 109 00:05:36,019 --> 00:05:38,219 And we'll want to write that in the x position. 110 00:05:38,220 --> 00:05:42,130 I'll switch colors just to not be monotonous. 111 00:05:42,129 --> 00:05:49,629 So minus 11/2 x times 1 is minus 11/2 x. 112 00:05:49,629 --> 00:05:53,909 And then, minus 11/2 x times 2x, well, we 113 00:05:53,910 --> 00:05:55,080 should know that is. 114 00:05:55,079 --> 00:05:56,409 But you can multiply them out. 115 00:05:56,410 --> 00:05:58,220 It'll be minus 11x squared. 116 00:05:58,220 --> 00:06:01,310 117 00:06:01,310 --> 00:06:02,839 I think you see what we're doing. 118 00:06:02,839 --> 00:06:05,599 After every step we're canceling out the largest 119 00:06:05,600 --> 00:06:09,740 degree of the polynomial we're dividing into. 120 00:06:09,740 --> 00:06:10,269 Fair enough? 121 00:06:10,269 --> 00:06:14,569 Now let's subtract this expression from this. 122 00:06:14,569 --> 00:06:16,639 And we'll get kind of our new intermediary remainder. 123 00:06:16,639 --> 00:06:18,759 And maybe that'll be the full remainder. 124 00:06:18,759 --> 00:06:19,599 So let's see. 125 00:06:19,600 --> 00:06:21,740 Minus 11x squared minus 11x squared. 126 00:06:21,740 --> 00:06:24,000 That's 0, so we don't have to write anything there. 127 00:06:24,000 --> 00:06:29,790 10x minus negative 11/2 x. 128 00:06:29,790 --> 00:06:33,069 Remember, we're subtracting this negative number from 10x. 129 00:06:33,069 --> 00:06:35,939 So if you're subtracting a negative number it's like 130 00:06:35,939 --> 00:06:38,129 adding a positive number. 131 00:06:38,129 --> 00:06:42,329 So you could view this as 10 plus 11/2. 132 00:06:42,329 --> 00:06:46,889 So 10 plus 11/2, that's 20/2 plus 11/2. 133 00:06:46,889 --> 00:06:49,029 That's 31/2 or 15.5. 134 00:06:49,029 --> 00:06:51,939 I'll just write 31/2 x. 135 00:06:51,939 --> 00:06:54,550 31/2 x. 136 00:06:54,550 --> 00:06:56,829 And then you could say that there was a 0 here and 137 00:06:56,829 --> 00:07:01,189 when you subtract 0 from minus 5 you get minus 5. 138 00:07:01,189 --> 00:07:07,420 And now we say, how many times does 2x go into 31/2 x. 139 00:07:07,420 --> 00:07:08,920 Let's do a little work on the side here. 140 00:07:08,920 --> 00:07:15,900 So if I have 31/2 x divided by 2x. 141 00:07:15,899 --> 00:07:17,784 Well, the x's will just cancel out. 142 00:07:17,785 --> 00:07:21,560 143 00:07:21,560 --> 00:07:24,350 This is equal to 31/4. 144 00:07:24,350 --> 00:07:26,730 This is the same thing as 31 over 2 times 1/2. 145 00:07:26,730 --> 00:07:28,420 So it's 31/4. 146 00:07:28,420 --> 00:07:31,810 So 2x goes into this expression 31/4 times and I'll 147 00:07:31,810 --> 00:07:33,596 switch colors. 148 00:07:33,596 --> 00:07:35,990 I'll switch to green. 149 00:07:35,990 --> 00:07:37,060 And that's a positive, right? 150 00:07:37,060 --> 00:07:38,470 You're dividing a positive into a positive. 151 00:07:38,470 --> 00:07:41,700 So plus 31/4 times. 152 00:07:41,699 --> 00:07:43,719 And I'm writing that in the-- you could view that in the 153 00:07:43,720 --> 00:07:46,830 constant space or the x to the 0 space. 154 00:07:46,829 --> 00:07:48,680 Or the 1 space even. 155 00:07:48,680 --> 00:07:51,069 So it goes into it 31/4 times. 156 00:07:51,069 --> 00:07:59,230 31/4 times 1 is 31/4. 157 00:07:59,230 --> 00:08:03,660 And 31/4 times 2x is 31/2 x. 158 00:08:03,660 --> 00:08:09,740 159 00:08:09,740 --> 00:08:11,689 And now we subtract. 160 00:08:11,689 --> 00:08:12,850 This is a plus here. 161 00:08:12,850 --> 00:08:15,340 We subtract the green expression from the light 162 00:08:15,339 --> 00:08:17,750 blue expression and we're left with this. 163 00:08:17,750 --> 00:08:20,110 When you subtract this from this you're left with 0, so 164 00:08:20,110 --> 00:08:21,319 nothing shows up there. 165 00:08:21,319 --> 00:08:26,279 And we're left with minus 5 minus 31/4. 166 00:08:26,279 --> 00:08:28,469 And we can just do a little bit of fraction work here. 167 00:08:28,470 --> 00:08:30,090 That's equal to, let's see. 168 00:08:30,089 --> 00:08:35,519 Minus 5 over 4 is minus 20 minus 31. 169 00:08:35,519 --> 00:08:37,699 All of that over 4. 170 00:08:37,700 --> 00:08:39,860 So that is equal to what? 171 00:08:39,860 --> 00:08:44,440 Minus 20, that's equal to minus 51. 172 00:08:44,440 --> 00:08:47,410 Minus 51/4. 173 00:08:47,409 --> 00:08:53,990 So, our answer is 2x plus 1 goes into 8x to the third minus 174 00:08:53,990 --> 00:08:59,139 7x squared plus 10x minus 5-- it goes into it 4x squared 175 00:08:59,139 --> 00:09:03,100 minus 11/2 x plus 31/4 times. 176 00:09:03,100 --> 00:09:07,320 But there is a remainder, and this is the remainder. 177 00:09:07,320 --> 00:09:09,460 And so a way to visualize this or another way to think about 178 00:09:09,460 --> 00:09:13,860 this problem so it's actually useful when we're actually 179 00:09:13,860 --> 00:09:15,269 solving real problems. 180 00:09:15,269 --> 00:09:17,319 And that you just don't view this as some kind of mechanical 181 00:09:17,320 --> 00:09:20,650 way to get problems right on a test that only tests 182 00:09:20,649 --> 00:09:22,889 algebraic long division. 183 00:09:22,889 --> 00:09:25,389 As another way to write this relationship you could write 184 00:09:25,389 --> 00:09:28,389 that-- let me do it in another color. 185 00:09:28,389 --> 00:09:30,819 I've used many of my colors already. 186 00:09:30,820 --> 00:09:40,800 So you can write that 2x plus 1 times this-- 4x squared. 187 00:09:40,799 --> 00:09:43,000 That's an x. 188 00:09:43,000 --> 00:09:49,950 Minus 11/2 x plus 31/4. 189 00:09:49,950 --> 00:09:52,160 Plus the remainder. 190 00:09:52,159 --> 00:09:54,579 So when you multiply these two out, and then if you were to 191 00:09:54,580 --> 00:09:59,980 add the remainder-- 51/4-- that that would equal-- and let 192 00:09:59,980 --> 00:10:00,879 me draw a dividing line. 193 00:10:00,879 --> 00:10:03,720 I don't want to confuse you with all this stuff here. 194 00:10:03,720 --> 00:10:06,379 That would equal this. 195 00:10:06,379 --> 00:10:11,700 That would equal 8x to the third minus 7x squared 196 00:10:11,700 --> 00:10:16,129 plus 10x minus 5. 197 00:10:16,129 --> 00:10:18,429 Anyway, I hope that helps. 198 00:10:18,429 --> 00:10:20,500 See you in the next video. 199 00:10:20,500 --> 00:10:20,878