1 00:00:01,360 --> 00:00:05,752 An algebraic fraction is one where the numerator and 2 00:00:05,752 --> 00:00:07,704 denominator, both polynomial expressions. 3 00:00:08,860 --> 00:00:12,880 This is an expression where every term is a multiple of a 4 00:00:12,880 --> 00:00:14,220 power of X, like. 5 00:00:14,830 --> 00:00:21,422 5X to the 4th plus 6X cubed plus 6 00:00:21,422 --> 00:00:23,894 7X plus 4. 7 00:00:24,860 --> 00:00:28,809 The degree of a polynomial is the power of the highest 8 00:00:28,809 --> 00:00:32,040 Terminix. So this is a polynomial of degree 4. 9 00:00:33,400 --> 00:00:37,950 The number in front of X in each case is the coefficient of that 10 00:00:37,950 --> 00:00:42,500 term. So the coefficient of X to the 4th is 5. The coefficient of 11 00:00:42,500 --> 00:00:43,800 X cubed is 6. 12 00:00:45,120 --> 00:00:52,700 Now look at these fractions X over X squared +2. 13 00:00:53,350 --> 00:00:59,587 Or X cubed plus three over X to the 4th plus 14 00:00:59,587 --> 00:01:01,855 X squared plus one. 15 00:01:02,900 --> 00:01:04,349 In both cases. 16 00:01:04,980 --> 00:01:07,212 The numerator is a polynomial of 17 00:01:07,212 --> 00:01:09,380 lower degree. Then the 18 00:01:09,380 --> 00:01:16,050 denominator. X's against X squared X cubed as against X to 19 00:01:16,050 --> 00:01:19,296 the 4th week. All these proper 20 00:01:19,296 --> 00:01:24,890 fractions. With other fractions, the polynomial may be of higher 21 00:01:24,890 --> 00:01:30,490 degree in the numerator. For instance, X fourth plus X 22 00:01:30,490 --> 00:01:32,170 squared plus X. 23 00:01:33,170 --> 00:01:41,010 Over X cubed plus X +2 or it may be of the same degree. 24 00:01:41,680 --> 00:01:49,084 Such as X plus four over X plus three. We call these 25 00:01:49,084 --> 00:01:55,486 improper fractions. Down, look like to look at 26 00:01:55,486 --> 00:02:02,470 how we add and subtract fractions. Take for instance 27 00:02:02,470 --> 00:02:04,798 these two fractions. 28 00:02:06,870 --> 00:02:14,163 In order to add these two fractions together, we need to 29 00:02:14,163 --> 00:02:19,467 find the lowest common denominator. In this particular 30 00:02:19,467 --> 00:02:27,423 case it's X minus 3 * 2 X plus one, so we 31 00:02:27,423 --> 00:02:31,401 say that this sum is the 32 00:02:31,401 --> 00:02:36,106 equivalent of. In the denominators we are going to 33 00:02:36,106 --> 00:02:39,328 have X minus 3 * 2 X plus one. 34 00:02:42,200 --> 00:02:49,350 In order to get from there to there, we multiplied by 2X plus 35 00:02:49,350 --> 00:02:54,850 one, we've multiplied the denominator by 2X plus one. So 36 00:02:54,850 --> 00:02:59,800 we must multiply the numerator by 2X plus one. 37 00:02:59,880 --> 00:03:05,532 And in order to get from here to here, we've multiplied the 38 00:03:05,532 --> 00:03:10,713 denominator by X minus three. So we've got to multiply the 39 00:03:10,713 --> 00:03:16,365 numerator by X minus three, and this gives us just X minus 40 00:03:16,365 --> 00:03:18,355 three. Now we need to collect 41 00:03:18,355 --> 00:03:23,310 up. The denominators of the same, so we can just right. 42 00:03:23,950 --> 00:03:31,342 X minus 3 * 2 X Plus One and on top we have 2 43 00:03:31,342 --> 00:03:36,622 * 2 X is 4X Minus, X gives us 3X. 44 00:03:38,260 --> 00:03:41,167 And we also have 2 * 1 is 2. 45 00:03:42,320 --> 00:03:49,250 Minus minus three is plus three, so 2 + 3 is +5 and that 46 00:03:49,250 --> 00:03:52,220 is the answer to that some. 47 00:03:53,010 --> 00:04:00,204 Sometimes in mathematics we need to do this operation in reverse. 48 00:04:00,204 --> 00:04:06,744 In calculus, for instance, or when dealing with the binomial 49 00:04:06,744 --> 00:04:11,560 theorem. We sometimes need to split a fraction up into its 50 00:04:11,560 --> 00:04:14,200 component parts, which are called partial fractions. Let's 51 00:04:14,200 --> 00:04:18,160 take the sum that I've just dealt with. We got the answer. 52 00:04:18,700 --> 00:04:21,280 Three X +5. 53 00:04:21,800 --> 00:04:22,490 Over. 54 00:04:24,410 --> 00:04:29,978 X minus 3 * 2 X plus one. 55 00:04:30,590 --> 00:04:35,198 So how do we get this back to its component parts? Well? 56 00:04:36,120 --> 00:04:40,560 We only have two factors in the denominator, X minus three and 57 00:04:40,560 --> 00:04:41,670 2X plus one. 58 00:04:42,870 --> 00:04:49,508 So. It must be something over X minus three plus 59 00:04:49,508 --> 00:04:52,840 something. Over 2X plus one. 60 00:04:53,900 --> 00:04:59,144 And what are these some things? They can only be plain numbers, 61 00:04:59,144 --> 00:05:04,825 because if they involved X or powers of X then these would be 62 00:05:04,825 --> 00:05:09,195 improper fractions, so we're quite entitled to say that 3X 63 00:05:09,195 --> 00:05:16,624 plus five over X minus 3 * 2 X Plus one is a over X minus three 64 00:05:16,624 --> 00:05:21,868 plus B over 2X plus one where A&B are just plain numbers. 65 00:05:22,800 --> 00:05:26,936 The next thing to do is to multiply everything through by 66 00:05:26,936 --> 00:05:31,448 what's on the bottom X minus 3 * 2 X plus one. 67 00:05:32,050 --> 00:05:37,042 If you multiply the left hand side by that, we just get three 68 00:05:37,042 --> 00:05:38,194 X +5 equals. 69 00:05:39,240 --> 00:05:46,320 A over X minus three times X minus 3 * 2 X plus one the 70 00:05:46,320 --> 00:05:51,512 X minus threes will cancel, and we're just left with a 71 00:05:51,512 --> 00:05:53,400 Times 2X plus one. 72 00:05:55,720 --> 00:06:02,707 B over 2X Plus One Times X minus 3 * 2 X Plus one. This time the 73 00:06:02,707 --> 00:06:08,461 2X plus ones will cancel and we just left with B Times X minus 74 00:06:08,461 --> 00:06:14,752 three. Now this is an identity, which means that it is true for 75 00:06:14,752 --> 00:06:16,516 all values of X. 76 00:06:17,340 --> 00:06:20,736 If this is so, then we can substitute special values for X 77 00:06:20,736 --> 00:06:22,434 and it will still be true. 78 00:06:23,470 --> 00:06:27,160 For instance, if we make X equal to minus 1/2. 79 00:06:28,330 --> 00:06:31,363 This bracket will become zero and a will disappear. 80 00:06:32,440 --> 00:06:36,268 If we make X equal to three, this bracket will become zero 81 00:06:36,268 --> 00:06:37,544 and be will disappear. 82 00:06:38,260 --> 00:06:39,884 And I'm going to do just that. 83 00:06:40,590 --> 00:06:43,910 If X equals minus 1/2. 84 00:06:44,520 --> 00:06:51,492 We get three times minus 1/2 is minus three over 2 + 85 00:06:51,492 --> 00:06:54,130 5. That is 0. 86 00:06:55,200 --> 00:07:02,088 Equals B times minus 1/2 - 3. 87 00:07:03,040 --> 00:07:09,868 This is just Seven over 2 and we get 7 over 2 88 00:07:09,868 --> 00:07:16,696 equals. This is minus 7 over 2 - 7 over 2B so 89 00:07:16,696 --> 00:07:20,110 B is equal to minus one. 90 00:07:22,990 --> 00:07:29,742 All right, this line in again 3X plus 91 00:07:29,742 --> 00:07:35,668 5. Equals a Times 2X plus one. 92 00:07:36,280 --> 00:07:38,770 Plus B times. 93 00:07:40,520 --> 00:07:42,749 X minus three. 94 00:07:43,330 --> 00:07:47,314 This time I want to try and find a, so I'm going to put X equal 95 00:07:47,314 --> 00:07:50,711 to 3. If X equals 96 00:07:50,711 --> 00:07:56,860 3. We have 3 threes and 9 + 5 is 14. 97 00:07:57,790 --> 00:08:00,380 3266 plus One is 7. 98 00:08:02,190 --> 00:08:07,740 That is going to be 0, so be will disappear, so A is equal to 99 00:08:07,740 --> 00:08:10,700 14 / 7. In other words, a IS2. 100 00:08:11,940 --> 00:08:15,564 We already had the equal to minus one. 101 00:08:16,820 --> 00:08:19,058 So what do we have now? 102 00:08:20,590 --> 00:08:24,376 I had three X +5 over. 103 00:08:25,510 --> 00:08:27,748 X minus three. 104 00:08:28,640 --> 00:08:34,540 2X plus one times 2X plus one equals a over. 105 00:08:35,140 --> 00:08:42,742 X minus three plus B over 2X plus one. Since A is 2 and 106 00:08:42,742 --> 00:08:50,344 B is minus one, we can see that this is 2 over X minus 107 00:08:50,344 --> 00:08:52,516 three plus, sorry minus. 108 00:08:53,410 --> 00:08:58,545 One over 2X Plus One, which is the sum that we started with 109 00:08:58,545 --> 00:09:02,890 and we have now broken this back into its component parts 110 00:09:02,890 --> 00:09:04,075 called partial fractions. 111 00:09:05,290 --> 00:09:12,710 Do another example. Let's say that we have to express 112 00:09:12,710 --> 00:09:20,130 3X over X minus one times X +2 in partial 113 00:09:20,130 --> 00:09:26,440 fractions. Again, we look at the denominator. The factors in the 114 00:09:26,440 --> 00:09:32,212 denominator X minus one and X +2. So we say that this 115 00:09:32,212 --> 00:09:38,946 expression is equal to a over X minus one plus B over X +2. 116 00:09:40,480 --> 00:09:45,996 We multiply through by X minus one times X +2 on the left hand 117 00:09:45,996 --> 00:09:48,360 side. This just gives us 3X. 118 00:09:50,440 --> 00:09:56,558 On the right hand side, a over X minus one times X minus one 119 00:09:56,558 --> 00:10:02,676 times X +2 X minus ones cancel out, and we're left with a Times 120 00:10:02,676 --> 00:10:09,900 X +2. Be over X +2 times X minus 1X Plus 2X 121 00:10:09,900 --> 00:10:16,489 plus Two's cancel out and we're left with B Times X 122 00:10:16,489 --> 00:10:17,687 minus one. 123 00:10:21,080 --> 00:10:25,160 This time the special values that I'm going to take our X 124 00:10:25,160 --> 00:10:28,900 equals minus two because that will make that zero and thus 125 00:10:28,900 --> 00:10:32,980 eliminate A and X equals 1, which will make that zero and 126 00:10:32,980 --> 00:10:34,000 thus eliminate B. 127 00:10:35,030 --> 00:10:38,246 If X equals minus 128 00:10:38,246 --> 00:10:44,828 2. We get three times minus two is minus 6. 129 00:10:45,630 --> 00:10:48,036 That is 0, so a disappears. 130 00:10:49,120 --> 00:10:55,444 Minus 2 - 1 is minus three, so this is minus 3B. 131 00:10:56,070 --> 00:10:56,790 So. 132 00:10:57,810 --> 00:11:04,950 B equals minus 6 divided by minus 3 equals 2. 133 00:11:06,130 --> 00:11:12,794 Alright, this expression in 134 00:11:12,794 --> 00:11:14,460 again. 135 00:11:16,110 --> 00:11:23,460 This time I'm going to put 136 00:11:23,460 --> 00:11:27,135 X equal to 137 00:11:27,135 --> 00:11:30,690 1. 3 * 1 138 00:11:30,690 --> 00:11:37,736 is 3. 1 + 2 is 3, so we get 3A. 139 00:11:38,350 --> 00:11:41,414 1 - 1 is 0 so be disappears. 140 00:11:42,550 --> 00:11:48,179 If 3A equals 3, then a is going to equal 1, so we've 141 00:11:48,179 --> 00:11:52,942 got a equal 1. We already had B equal to two. 142 00:11:54,380 --> 00:11:56,151 I'm not going to write the whole 143 00:11:56,151 --> 00:11:59,360 expression in again. We have 3X. 144 00:11:59,870 --> 00:12:07,550 Over X minus one times X +2 equals. 145 00:12:08,050 --> 00:12:15,505 One over X minus one because a is 1 + 2 over X +2 because 146 00:12:15,505 --> 00:12:19,481 be is 2 and that is the answer. 147 00:12:20,290 --> 00:12:28,154 Sometimes the denominators more awkward, for example, to 148 00:12:28,154 --> 00:12:36,018 express 3X plus one over X minus one 149 00:12:36,018 --> 00:12:39,950 squared times X +2. 150 00:12:40,900 --> 00:12:43,900 There are actually three possibilities for a denominator 151 00:12:43,900 --> 00:12:45,400 in the partial fraction. 152 00:12:45,930 --> 00:12:50,909 We've got X minus One X +2, but there's also the possibility of 153 00:12:50,909 --> 00:12:52,441 X minus 1 squared. 154 00:12:53,550 --> 00:13:01,518 So we write down a over X minus one plus B over 155 00:13:01,518 --> 00:13:04,174 X minus 1 squared. 156 00:13:04,690 --> 00:13:08,700 Plus C over X +2. 157 00:13:10,490 --> 00:13:16,814 Again, we multiply through by the bottom line here, so we get 158 00:13:16,814 --> 00:13:23,665 a over X minus one times X minus one squared times X +2. 159 00:13:23,665 --> 00:13:30,516 One of the X minus ones will cancel, leaving us with 3X plus 160 00:13:30,516 --> 00:13:35,786 one equals a Times X minus one times X +2. 161 00:13:36,690 --> 00:13:43,256 B over X minus one squared times X minus one squared times X +2. 162 00:13:43,256 --> 00:13:49,353 Both of the X minus one squared will cancel, leaving us with B 163 00:13:49,353 --> 00:13:50,760 Times X +2. 164 00:13:51,700 --> 00:13:57,454 And then we have C over X +2 times X minus one squared times 165 00:13:57,454 --> 00:14:03,619 X +2. This time the X +2 is will cancel, leaving us with C Times 166 00:14:03,619 --> 00:14:05,263 X minus 1 squared. 167 00:14:07,090 --> 00:14:12,326 Again, the special values X equals one will make this zero, 168 00:14:12,326 --> 00:14:18,514 so a will disappear and it will make this zero. So see will 169 00:14:18,514 --> 00:14:24,041 disappear. If X equals one, we have 3X Plus One is 4. 170 00:14:24,650 --> 00:14:27,050 That zero so that expression 171 00:14:27,050 --> 00:14:32,080 disappears. 1 + 2 is 3, so we have 3B. 172 00:14:33,600 --> 00:14:40,544 This is 0, so this disappears. So we have 4 equals 3B. Giving B 173 00:14:40,544 --> 00:14:42,528 equals 4 over 3. 174 00:14:44,310 --> 00:14:47,268 If X equals. 175 00:14:47,990 --> 00:14:49,200 Minus 2. 176 00:14:51,390 --> 00:14:56,766 We have minus 2 * 3 is minus 6 Plus One is minus 5. 177 00:14:57,350 --> 00:15:00,542 Equals this is 0, so this 178 00:15:00,542 --> 00:15:05,398 disappears. This is 0, so this disappears minus 2. 179 00:15:05,990 --> 00:15:13,172 Minus one is minus 3 squared is 9, so we have minus five is 180 00:15:13,172 --> 00:15:16,763 9C, which gives us C is minus 181 00:15:16,763 --> 00:15:18,690 5. Over 9. 182 00:15:20,860 --> 00:15:28,324 We now need to find a. I'm just going to write this 183 00:15:28,324 --> 00:15:30,190 expression out again. 184 00:15:30,210 --> 00:15:43,062 I've written 185 00:15:43,062 --> 00:15:51,454 the. Expression following, see out 186 00:15:51,454 --> 00:15:55,486 like that because in a minute I'm going to multiply it out. 187 00:15:56,480 --> 00:16:00,062 Unfortunately, there's no special value of X that will 188 00:16:00,062 --> 00:16:02,450 eliminate B&C. To give us A. 189 00:16:03,130 --> 00:16:07,400 We can use any special value. We could use X equals 0. This would 190 00:16:07,400 --> 00:16:11,365 give us an equation in AB&C since we already know be in. See 191 00:16:11,365 --> 00:16:12,890 this would give us a. 192 00:16:13,590 --> 00:16:16,983 But I'm going to use a different technique, one 193 00:16:16,983 --> 00:16:19,245 called equating coefficients, and to do 194 00:16:19,245 --> 00:16:22,638 that I've got to multiply this lot right out. 195 00:16:24,330 --> 00:16:27,520 So we get equals a. 196 00:16:28,570 --> 00:16:31,486 And we have an X Times X for X 197 00:16:31,486 --> 00:16:37,294 squared. We have a minus 1X plus 2X, so that gives us Plus X. 198 00:16:38,080 --> 00:16:42,076 And we have minus 1 * 2 which gives us minus 2. 199 00:16:42,810 --> 00:16:46,082 And then plus BX 200 00:16:46,082 --> 00:16:49,390 +2. Plus C. 201 00:16:49,980 --> 00:16:55,860 X times X is X squared. We have a minus X under minus six, so 202 00:16:55,860 --> 00:17:00,956 that's minus 2X and then minus one times minus one is plus one. 203 00:17:02,250 --> 00:17:07,593 I'm not going to collect up all the terms. For instance, we have 204 00:17:07,593 --> 00:17:13,758 an A Times X squared here and we have a C Times X squared here. 205 00:17:13,758 --> 00:17:17,046 So we have a plus C Times X 206 00:17:17,046 --> 00:17:24,410 squared altogether. We also have an A Times XAB Times X&A minus 207 00:17:24,410 --> 00:17:26,690 two C Times X. 208 00:17:28,000 --> 00:17:34,180 A+B minus two C Times X? 209 00:17:34,730 --> 00:17:37,526 And finally we have minus 2A. 210 00:17:38,590 --> 00:17:40,738 2B and C. 211 00:17:41,260 --> 00:17:49,040 So the constant becomes minus 2A plus 2B Plus C. 212 00:17:51,380 --> 00:17:55,066 Now. So we have 3X plus one 213 00:17:55,066 --> 00:17:56,730 equals. This line. 214 00:17:57,900 --> 00:18:00,042 But in this line, we have a 215 00:18:00,042 --> 00:18:03,265 Turman X squared. 3X plus one doesn't have 216 00:18:03,265 --> 00:18:04,533 anything in X squared. 217 00:18:05,650 --> 00:18:10,900 But this is an identity. It must be true for all values of X, and 218 00:18:10,900 --> 00:18:16,850 the only way that this can be true is for A plus E to be 0 so 219 00:18:16,850 --> 00:18:21,400 that X squared disappears on this side. So we can say that a 220 00:18:21,400 --> 00:18:22,800 plus C equals 0. 221 00:18:23,750 --> 00:18:28,870 We already know that C is minus 5 over 9, so in order for 8 plus 222 00:18:28,870 --> 00:18:30,150 C to be 0. 223 00:18:30,720 --> 00:18:33,324 A must be plus five over 9. 224 00:18:33,870 --> 00:18:37,420 And we already worked out B as being equal to. 225 00:18:37,940 --> 00:18:42,802 For over 3, this means that we can write out the solution to 226 00:18:42,802 --> 00:18:43,924 the whole problem. 227 00:18:44,480 --> 00:18:51,704 3X plus one over X minus one squared 228 00:18:51,704 --> 00:18:54,413 times X +2. 229 00:18:55,000 --> 00:19:02,346 Equals. A5 over 9X minus one plus B is 230 00:19:02,346 --> 00:19:09,024 4 over 3 four over 3X minus 1 squared. 231 00:19:09,760 --> 00:19:16,774 See is minus 5 over 9, so we have minus five over 9 X 232 00:19:16,774 --> 00:19:23,000 +2. Another case we must consider. 233 00:19:23,540 --> 00:19:27,986 Is where the denominator contains a quadratic that can't 234 00:19:27,986 --> 00:19:30,950 be factorized as in 5X over. 235 00:19:31,460 --> 00:19:37,800 X squared plus X Plus One Times X minus 2. 236 00:19:38,500 --> 00:19:42,031 If we to express this in partial fractions, the two denominators 237 00:19:42,031 --> 00:19:46,525 are going to be X squared plus X Plus One and X minus 2. 238 00:19:47,120 --> 00:19:52,076 When the denominator is X squared plus 6 plus one, we have 239 00:19:52,076 --> 00:19:56,206 to consider the possibility that the numerator can contain a 240 00:19:56,206 --> 00:19:59,923 termine ex, because the numerator would still be of 241 00:19:59,923 --> 00:20:03,227 lower degree than the denominator, and this would 242 00:20:03,227 --> 00:20:08,596 still therefore be a proper fraction. So we write a X plus B 243 00:20:08,596 --> 00:20:11,487 over X squared plus X plus one. 244 00:20:12,090 --> 00:20:19,158 Plus C over X minus two as before. We multiply this out 245 00:20:19,158 --> 00:20:26,226 so we get that five X equals X plus B Times X 246 00:20:26,226 --> 00:20:33,454 minus 2. Plus C Times 247 00:20:33,454 --> 00:20:40,010 X squared. Plus 6 + 1. 248 00:20:41,630 --> 00:20:43,919 One special value we can use is 249 00:20:43,919 --> 00:20:46,510 X equals 2. And if. 250 00:20:47,020 --> 00:20:50,744 X equals 2, we 251 00:20:50,744 --> 00:20:57,005 get 5X5210. This is 0, so this all disappears and we 252 00:20:57,005 --> 00:21:04,115 get 2 twos of 4 + 2 is 6 plus one is 7, so 10 equals 7 C. 253 00:21:04,940 --> 00:21:08,516 Giving C equals 10 over 7. 254 00:21:09,520 --> 00:21:14,200 Unfortunately, there's no value for X would enable us to get rid 255 00:21:14,200 --> 00:21:19,270 of C, so we're going to have to use the technique of equating 256 00:21:19,270 --> 00:21:25,770 coefficients. I'll write this out again. 257 00:21:26,450 --> 00:21:32,430 In order to equate 258 00:21:32,430 --> 00:21:38,410 coefficients, I'm going to 259 00:21:38,410 --> 00:21:44,390 have to multiply this 260 00:21:44,390 --> 00:21:45,885 out. 261 00:21:46,900 --> 00:21:50,026 X times X is X squared. 262 00:21:51,140 --> 00:21:54,794 X times minus two is minus two 263 00:21:54,794 --> 00:22:01,640 AX. B times X is BXB times minus two 264 00:22:01,640 --> 00:22:08,740 gives us minus 2B Plus CX squared Plus CX Plus 265 00:22:08,740 --> 00:22:13,806 C. Again, I'm going to collect like terms. So for instance for 266 00:22:13,806 --> 00:22:15,190 X squared we have. 267 00:22:16,380 --> 00:22:23,090 AX squared and CX squared. So we have a plus 268 00:22:23,090 --> 00:22:29,800 CX squared for X. We have a minus two AAB&C. 269 00:22:30,310 --> 00:22:38,200 So minus two A+B Plus CX and for a constant 270 00:22:38,200 --> 00:22:42,145 we have minus 2B Plus 271 00:22:42,145 --> 00:22:48,475 C. We still need to find 272 00:22:48,475 --> 00:22:50,609 both A&B. 273 00:22:51,870 --> 00:22:55,734 For two unknowns we need 2 equations, so we are going to 274 00:22:55,734 --> 00:22:57,666 have to solve for two different 275 00:22:57,666 --> 00:23:02,438 coefficients. Now the left hand side is just 5X, so there is no 276 00:23:02,438 --> 00:23:03,674 coefficient in X squared. 277 00:23:04,280 --> 00:23:09,936 In order to eliminate X squared, we can say that a plus C equals 278 00:23:09,936 --> 00:23:17,465 0. We already know what see is 10 over 7. In order for a plus C 279 00:23:17,465 --> 00:23:22,008 to be 0, this will make a minus 10 over 7. 280 00:23:24,280 --> 00:23:28,700 The left hand side also has no constant coefficient, so 281 00:23:28,700 --> 00:23:33,120 that means that this expression must be 0. So we 282 00:23:33,120 --> 00:23:36,214 say minus 2B Plus C equals 0. 283 00:23:37,620 --> 00:23:40,876 Giving us. C equals 284 00:23:40,876 --> 00:23:48,100 2B. Or B equals C over two, which gives us B as being. 285 00:23:48,610 --> 00:23:51,079 5 over 7. 286 00:23:52,960 --> 00:24:00,352 So we have a equal to minus 10 over 7B equal to 287 00:24:00,352 --> 00:24:06,512 five over 7 and C equal to 10 over 7. 288 00:24:08,000 --> 00:24:11,200 This means that 5X over. 289 00:24:11,700 --> 00:24:15,126 X squared plus X plus one. 290 00:24:15,670 --> 00:24:23,090 Times X minus two is equal to a X which is minus 10 over 291 00:24:23,090 --> 00:24:30,932 7X. Plus B, which is 5 over 7 all over X squared plus 292 00:24:30,932 --> 00:24:32,585 X plus one. 293 00:24:33,780 --> 00:24:40,892 Plus C, which is 10 over 7 over X minus two and are now 294 00:24:40,892 --> 00:24:47,496 tidy. This up the Seven comes down to be multiplied by the X 295 00:24:47,496 --> 00:24:54,608 squared plus X plus one. So we get minus 10X plus five over 7 296 00:24:54,608 --> 00:25:01,212 X squared plus X Plus One plus and again the Seven comes down 297 00:25:01,212 --> 00:25:03,752 10 over 7X minus 2. 298 00:25:03,860 --> 00:25:10,202 Equals and to finish it off we need to take five out of this 299 00:25:10,202 --> 00:25:15,185 expression as a factor, which gives us five times minus 2X 300 00:25:15,185 --> 00:25:22,433 plus one over 7 X squared plus X plus 1 + 10 over 7X minus 2. 301 00:25:23,300 --> 00:25:29,160 So far I've only dealt with proper fractions where the 302 00:25:29,160 --> 00:25:35,020 numerator is of lower degree than the denominator. Now, like 303 00:25:35,020 --> 00:25:38,536 to look at an improper fraction. 304 00:25:39,220 --> 00:25:40,950 Let's Express. 305 00:25:42,170 --> 00:25:49,202 4X cubed plus 10X plus four over X 306 00:25:49,202 --> 00:25:52,718 into 2X plus one. 307 00:25:53,390 --> 00:25:55,358 In partial fractions. 308 00:25:57,450 --> 00:25:59,460 The numerator is of degree 3. 309 00:26:00,590 --> 00:26:05,105 The denominator, if you multiply the X by the two X, you get 2 X 310 00:26:05,105 --> 00:26:06,610 squared, so the denominator is 311 00:26:06,610 --> 00:26:10,576 of degree 2. This means that this is an improper fraction. 312 00:26:11,750 --> 00:26:15,998 What this means is that if you divide the numerator by the 313 00:26:15,998 --> 00:26:19,892 denominator, you're going to be dividing otermin X cubed by a 314 00:26:19,892 --> 00:26:21,308 term in X squared. 315 00:26:22,180 --> 00:26:24,640 So you could get a Terminix. 316 00:26:25,160 --> 00:26:29,788 Which means that we have to write down acts. We may also get 317 00:26:29,788 --> 00:26:33,348 a constant term, so we have to write down B. 318 00:26:33,910 --> 00:26:36,280 Then we can do our fractions. 319 00:26:37,130 --> 00:26:44,510 If I now multiply but through I get a X 320 00:26:44,510 --> 00:26:51,890 Times X Times 2X plus one, so we get 4X 321 00:26:51,890 --> 00:26:55,580 cubed plus 10X plus four 322 00:26:55,580 --> 00:26:58,850 equals a. X squared 323 00:26:59,410 --> 00:27:02,558 Times 2X plus one. 324 00:27:02,560 --> 00:27:06,250 Plus BX times 2X 325 00:27:06,250 --> 00:27:13,206 plus one. Plus C Times 2X plus one. 326 00:27:14,500 --> 00:27:17,670 Plus DX 327 00:27:21,530 --> 00:27:23,039 Using special values. 328 00:27:23,730 --> 00:27:29,456 If I use X equals 0, then the term the D, the B, and 329 00:27:29,456 --> 00:27:33,955 the A are all going to disappear and I'm just left 330 00:27:33,955 --> 00:27:36,818 with see. So if X equals 0. 331 00:27:37,930 --> 00:27:42,675 X cubed is zero, X is zero. I just get 4 equal to. 332 00:27:44,370 --> 00:27:50,706 2X is 0, so it's just C, so we have C equal to four. The other 333 00:27:50,706 --> 00:27:53,082 special value is X equal to 334 00:27:53,082 --> 00:27:59,391 minus 1/2. If X equals minus Alpha, this is 0, so this will 335 00:27:59,391 --> 00:28:02,397 disappear. This is 0, so this 336 00:28:02,397 --> 00:28:06,696 will disappear. And this will disappear, just leaving me with 337 00:28:06,696 --> 00:28:09,320 D. So I get. 338 00:28:10,330 --> 00:28:15,263 Minus 1/2. Cubed is minus an eighth, so we 339 00:28:15,263 --> 00:28:17,508 get minus four over 8. 340 00:28:18,650 --> 00:28:25,922 Plus 10 times minus 1/2 inches minus 10 over 2 + 4 341 00:28:25,922 --> 00:28:29,380 equals. D times minus 342 00:28:29,380 --> 00:28:36,478 1/2. I'll just write that down again, 343 00:28:36,478 --> 00:28:44,182 minus four over 8 - 10 over 2. 344 00:28:45,480 --> 00:28:49,340 +4. Equals minus 345 00:28:49,340 --> 00:28:54,704 1/2 D. Minus 4 over 8 is just minus 1/2. 346 00:28:55,290 --> 00:29:02,440 Minus 10 over 2 is minus 5 + 4 equals minus half D. 347 00:29:04,290 --> 00:29:11,584 Minus 5 + 4 is minus one, so I've got minus 1 1/2 equals 348 00:29:11,584 --> 00:29:13,147 minus 1/2 D. 349 00:29:16,230 --> 00:29:21,433 Minus 1 1/2 is just three times minus 1/2, so this 350 00:29:21,433 --> 00:29:23,798 gives us D equal 3. 351 00:29:25,090 --> 00:29:31,054 Special values won't give me a or be, so I'm going to have to 352 00:29:31,054 --> 00:29:35,314 equate coefficients. This means I have to write this expression 353 00:29:35,314 --> 00:29:40,875 out again. 4X cubed plus 354 00:29:40,875 --> 00:29:48,198 10X. +4 equals a X squared times. 355 00:29:48,770 --> 00:29:55,218 2X plus one plus BX times 2X plus 356 00:29:55,218 --> 00:29:57,230 one. Plus 357 00:29:58,260 --> 00:30:03,860 C times 2X plus one plus DX. 358 00:30:05,550 --> 00:30:07,496 I'm now going to multiply this out. 359 00:30:08,940 --> 00:30:13,860 X squared times 2X is 2A X cubed. 360 00:30:14,920 --> 00:30:18,864 X squared times one is just X squared. 361 00:30:19,960 --> 00:30:22,438 This gives me 2B X squared. 362 00:30:24,850 --> 00:30:27,961 This gives me 363 00:30:27,961 --> 00:30:31,576 BX. This gives Me 2 364 00:30:31,576 --> 00:30:36,818 CX. This gives me C. 365 00:30:37,990 --> 00:30:38,660 And then. 366 00:30:39,840 --> 00:30:47,330 Plus DX And collecting terms, we only have one Turman X cubed, so 367 00:30:47,330 --> 00:30:50,330 that is just 2A X cubed. 368 00:30:51,010 --> 00:30:55,993 Plus we have two terms in X squared, A and 2B. 369 00:30:56,830 --> 00:31:04,550 We have three terms in XB2C and D. 370 00:31:05,120 --> 00:31:10,632 And finally, the constant 371 00:31:10,632 --> 00:31:13,388 term see. 372 00:31:15,050 --> 00:31:19,572 Now look at the Turman X cubed. We have 4X cubed on the left. 373 00:31:20,330 --> 00:31:25,860 And two AX cubed on the right. This means that 2A must be equal 374 00:31:25,860 --> 00:31:32,282 to 4. Giving us a equal to two now look at the Turman X 375 00:31:32,282 --> 00:31:35,850 squared. There is no Turman X squared on the left. 376 00:31:36,600 --> 00:31:39,228 And on the right we have a plus 2B. 377 00:31:40,310 --> 00:31:43,729 This means that as there isn't Turman X squared on the left, a 378 00:31:43,729 --> 00:31:45,570 plus 2B must be equal to 0. 379 00:31:46,120 --> 00:31:49,648 So we have a plus 2B 380 00:31:49,648 --> 00:31:52,820 equals 0. Which means that. 381 00:31:53,450 --> 00:31:56,951 A equals minus 382 00:31:56,951 --> 00:32:01,560 2B. Which means that B equals. 383 00:32:03,040 --> 00:32:07,779 Minus two over 2 equals minus one. 384 00:32:09,110 --> 00:32:15,088 I'll just write those values in again. 385 00:32:15,800 --> 00:32:17,768 A equals 2. 386 00:32:18,510 --> 00:32:24,936 B equals minus one C equals 4D equals 3. 387 00:32:27,700 --> 00:32:34,124 So if we take our original expression 4X cubed plus 10X 388 00:32:34,124 --> 00:32:37,044 plus four over X times. 389 00:32:38,890 --> 00:32:44,810 2X plus one. This is equal to axe, so 2X. 390 00:32:45,390 --> 00:32:52,431 Minus B. Plus see over X, so that's four over X Plus D 391 00:32:52,431 --> 00:32:57,282 over 2X Plus One which is 3 over 2X plus one.