WEBVTT 00:00:01.360 --> 00:00:05.752 An algebraic fraction is one where the numerator and 00:00:05.752 --> 00:00:07.704 denominator, both polynomial expressions. 00:00:08.860 --> 00:00:12.880 This is an expression where every term is a multiple of a 00:00:12.880 --> 00:00:14.220 power of X, like. 00:00:14.830 --> 00:00:21.422 5X to the 4th plus 6X cubed plus 00:00:21.422 --> 00:00:23.894 7X plus 4. 00:00:24.860 --> 00:00:28.809 The degree of a polynomial is the power of the highest 00:00:28.809 --> 00:00:32.040 Terminix. So this is a polynomial of degree 4. 00:00:33.400 --> 00:00:37.950 The number in front of X in each case is the coefficient of that 00:00:37.950 --> 00:00:42.500 term. So the coefficient of X to the 4th is 5. The coefficient of 00:00:42.500 --> 00:00:43.800 X cubed is 6. 00:00:45.120 --> 00:00:52.700 Now look at these fractions X over X squared +2. 00:00:53.350 --> 00:00:59.587 Or X cubed plus three over X to the 4th plus 00:00:59.587 --> 00:01:01.855 X squared plus one. 00:01:02.900 --> 00:01:04.349 In both cases. 00:01:04.980 --> 00:01:07.212 The numerator is a polynomial of 00:01:07.212 --> 00:01:09.380 lower degree. Then the 00:01:09.380 --> 00:01:16.050 denominator. X's against X squared X cubed as against X to 00:01:16.050 --> 00:01:19.296 the 4th week. All these proper 00:01:19.296 --> 00:01:24.890 fractions. With other fractions, the polynomial may be of higher 00:01:24.890 --> 00:01:30.490 degree in the numerator. For instance, X fourth plus X 00:01:30.490 --> 00:01:32.170 squared plus X. 00:01:33.170 --> 00:01:41.010 Over X cubed plus X +2 or it may be of the same degree. 00:01:41.680 --> 00:01:49.084 Such as X plus four over X plus three. We call these 00:01:49.084 --> 00:01:55.486 improper fractions. Down, look like to look at 00:01:55.486 --> 00:02:02.470 how we add and subtract fractions. Take for instance 00:02:02.470 --> 00:02:04.798 these two fractions. 00:02:06.870 --> 00:02:14.163 In order to add these two fractions together, we need to 00:02:14.163 --> 00:02:19.467 find the lowest common denominator. In this particular 00:02:19.467 --> 00:02:27.423 case it's X minus 3 * 2 X plus one, so we 00:02:27.423 --> 00:02:31.401 say that this sum is the 00:02:31.401 --> 00:02:36.106 equivalent of. In the denominators we are going to 00:02:36.106 --> 00:02:39.328 have X minus 3 * 2 X plus one. 00:02:42.200 --> 00:02:49.350 In order to get from there to there, we multiplied by 2X plus 00:02:49.350 --> 00:02:54.850 one, we've multiplied the denominator by 2X plus one. So 00:02:54.850 --> 00:02:59.800 we must multiply the numerator by 2X plus one. 00:02:59.880 --> 00:03:05.532 And in order to get from here to here, we've multiplied the 00:03:05.532 --> 00:03:10.713 denominator by X minus three. So we've got to multiply the 00:03:10.713 --> 00:03:16.365 numerator by X minus three, and this gives us just X minus 00:03:16.365 --> 00:03:18.355 three. Now we need to collect 00:03:18.355 --> 00:03:23.310 up. The denominators of the same, so we can just right. 00:03:23.950 --> 00:03:31.342 X minus 3 * 2 X Plus One and on top we have 2 00:03:31.342 --> 00:03:36.622 * 2 X is 4X Minus, X gives us 3X. 00:03:38.260 --> 00:03:41.167 And we also have 2 * 1 is 2. 00:03:42.320 --> 00:03:49.250 Minus minus three is plus three, so 2 + 3 is +5 and that 00:03:49.250 --> 00:03:52.220 is the answer to that some. 00:03:53.010 --> 00:04:00.204 Sometimes in mathematics we need to do this operation in reverse. 00:04:00.204 --> 00:04:06.744 In calculus, for instance, or when dealing with the binomial 00:04:06.744 --> 00:04:11.560 theorem. We sometimes need to split a fraction up into its 00:04:11.560 --> 00:04:14.200 component parts, which are called partial fractions. Let's 00:04:14.200 --> 00:04:18.160 take the sum that I've just dealt with. We got the answer. 00:04:18.700 --> 00:04:21.280 Three X +5. 00:04:21.800 --> 00:04:22.490 Over. 00:04:24.410 --> 00:04:29.978 X minus 3 * 2 X plus one. 00:04:30.590 --> 00:04:35.198 So how do we get this back to its component parts? Well? 00:04:36.120 --> 00:04:40.560 We only have two factors in the denominator, X minus three and 00:04:40.560 --> 00:04:41.670 2X plus one. 00:04:42.870 --> 00:04:49.508 So. It must be something over X minus three plus 00:04:49.508 --> 00:04:52.840 something. Over 2X plus one. 00:04:53.900 --> 00:04:59.144 And what are these some things? They can only be plain numbers, 00:04:59.144 --> 00:05:04.825 because if they involved X or powers of X then these would be 00:05:04.825 --> 00:05:09.195 improper fractions, so we're quite entitled to say that 3X 00:05:09.195 --> 00:05:16.624 plus five over X minus 3 * 2 X Plus one is a over X minus three 00:05:16.624 --> 00:05:21.868 plus B over 2X plus one where A&B are just plain numbers. 00:05:22.800 --> 00:05:26.936 The next thing to do is to multiply everything through by 00:05:26.936 --> 00:05:31.448 what's on the bottom X minus 3 * 2 X plus one. 00:05:32.050 --> 00:05:37.042 If you multiply the left hand side by that, we just get three 00:05:37.042 --> 00:05:38.194 X +5 equals. 00:05:39.240 --> 00:05:46.320 A over X minus three times X minus 3 * 2 X plus one the 00:05:46.320 --> 00:05:51.512 X minus threes will cancel, and we're just left with a 00:05:51.512 --> 00:05:53.400 Times 2X plus one. 00:05:55.720 --> 00:06:02.707 B over 2X Plus One Times X minus 3 * 2 X Plus one. This time the 00:06:02.707 --> 00:06:08.461 2X plus ones will cancel and we just left with B Times X minus 00:06:08.461 --> 00:06:14.752 three. Now this is an identity, which means that it is true for 00:06:14.752 --> 00:06:16.516 all values of X. 00:06:17.340 --> 00:06:20.736 If this is so, then we can substitute special values for X 00:06:20.736 --> 00:06:22.434 and it will still be true. 00:06:23.470 --> 00:06:27.160 For instance, if we make X equal to minus 1/2. 00:06:28.330 --> 00:06:31.363 This bracket will become zero and a will disappear. 00:06:32.440 --> 00:06:36.268 If we make X equal to three, this bracket will become zero 00:06:36.268 --> 00:06:37.544 and be will disappear. 00:06:38.260 --> 00:06:39.884 And I'm going to do just that. 00:06:40.590 --> 00:06:43.910 If X equals minus 1/2. 00:06:44.520 --> 00:06:51.492 We get three times minus 1/2 is minus three over 2 + 00:06:51.492 --> 00:06:54.130 5. That is 0. 00:06:55.200 --> 00:07:02.088 Equals B times minus 1/2 - 3. 00:07:03.040 --> 00:07:09.868 This is just Seven over 2 and we get 7 over 2 00:07:09.868 --> 00:07:16.696 equals. This is minus 7 over 2 - 7 over 2B so 00:07:16.696 --> 00:07:20.110 B is equal to minus one. 00:07:22.990 --> 00:07:29.742 All right, this line in again 3X plus 00:07:29.742 --> 00:07:35.668 5. Equals a Times 2X plus one. 00:07:36.280 --> 00:07:38.770 Plus B times. 00:07:40.520 --> 00:07:42.749 X minus three. 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 00:07:47.314 --> 00:07:50.711 to 3. If X equals 00:07:50.711 --> 00:07:56.860 3. We have 3 threes and 9 + 5 is 14. 00:07:57.790 --> 00:08:00.380 3266 plus One is 7. 00:08:02.190 --> 00:08:07.740 That is going to be 0, so be will disappear, so A is equal to 00:08:07.740 --> 00:08:10.700 14 / 7. In other words, a IS2. 00:08:11.940 --> 00:08:15.564 We already had the equal to minus one. 00:08:16.820 --> 00:08:19.058 So what do we have now? 00:08:20.590 --> 00:08:24.376 I had three X +5 over. 00:08:25.510 --> 00:08:27.748 X minus three. 00:08:28.640 --> 00:08:34.540 2X plus one times 2X plus one equals a over. 00:08:35.140 --> 00:08:42.742 X minus three plus B over 2X plus one. Since A is 2 and 00:08:42.742 --> 00:08:50.344 B is minus one, we can see that this is 2 over X minus 00:08:50.344 --> 00:08:52.516 three plus, sorry minus. 00:08:53.410 --> 00:08:58.545 One over 2X Plus One, which is the sum that we started with 00:08:58.545 --> 00:09:02.890 and we have now broken this back into its component parts 00:09:02.890 --> 00:09:04.075 called partial fractions. 00:09:05.290 --> 00:09:12.710 Do another example. Let's say that we have to express 00:09:12.710 --> 00:09:20.130 3X over X minus one times X +2 in partial 00:09:20.130 --> 00:09:26.440 fractions. Again, we look at the denominator. The factors in the 00:09:26.440 --> 00:09:32.212 denominator X minus one and X +2. So we say that this 00:09:32.212 --> 00:09:38.946 expression is equal to a over X minus one plus B over X +2. 00:09:40.480 --> 00:09:45.996 We multiply through by X minus one times X +2 on the left hand 00:09:45.996 --> 00:09:48.360 side. This just gives us 3X. 00:09:50.440 --> 00:09:56.558 On the right hand side, a over X minus one times X minus one 00:09:56.558 --> 00:10:02.676 times X +2 X minus ones cancel out, and we're left with a Times 00:10:02.676 --> 00:10:09.900 X +2. Be over X +2 times X minus 1X Plus 2X 00:10:09.900 --> 00:10:16.489 plus Two's cancel out and we're left with B Times X 00:10:16.489 --> 00:10:17.687 minus one. 00:10:21.080 --> 00:10:25.160 This time the special values that I'm going to take our X 00:10:25.160 --> 00:10:28.900 equals minus two because that will make that zero and thus 00:10:28.900 --> 00:10:32.980 eliminate A and X equals 1, which will make that zero and 00:10:32.980 --> 00:10:34.000 thus eliminate B. 00:10:35.030 --> 00:10:38.246 If X equals minus 00:10:38.246 --> 00:10:44.828 2. We get three times minus two is minus 6. 00:10:45.630 --> 00:10:48.036 That is 0, so a disappears. 00:10:49.120 --> 00:10:55.444 Minus 2 - 1 is minus three, so this is minus 3B. 00:10:56.070 --> 00:10:56.790 So. 00:10:57.810 --> 00:11:04.950 B equals minus 6 divided by minus 3 equals 2. 00:11:06.130 --> 00:11:12.794 Alright, this expression in 00:11:12.794 --> 00:11:14.460 again. 00:11:16.110 --> 00:11:23.460 This time I'm going to put 00:11:23.460 --> 00:11:27.135 X equal to 00:11:27.135 --> 00:11:30.690 1. 3 * 1 00:11:30.690 --> 00:11:37.736 is 3. 1 + 2 is 3, so we get 3A. 00:11:38.350 --> 00:11:41.414 1 - 1 is 0 so be disappears. 00:11:42.550 --> 00:11:48.179 If 3A equals 3, then a is going to equal 1, so we've 00:11:48.179 --> 00:11:52.942 got a equal 1. We already had B equal to two. 00:11:54.380 --> 00:11:56.151 I'm not going to write the whole 00:11:56.151 --> 00:11:59.360 expression in again. We have 3X. 00:11:59.870 --> 00:12:07.550 Over X minus one times X +2 equals. 00:12:08.050 --> 00:12:15.505 One over X minus one because a is 1 + 2 over X +2 because 00:12:15.505 --> 00:12:19.481 be is 2 and that is the answer. 00:12:20.290 --> 00:12:28.154 Sometimes the denominators more awkward, for example, to 00:12:28.154 --> 00:12:36.018 express 3X plus one over X minus one 00:12:36.018 --> 00:12:39.950 squared times X +2. 00:12:40.900 --> 00:12:43.900 There are actually three possibilities for a denominator 00:12:43.900 --> 00:12:45.400 in the partial fraction. 00:12:45.930 --> 00:12:50.909 We've got X minus One X +2, but there's also the possibility of 00:12:50.909 --> 00:12:52.441 X minus 1 squared. 00:12:53.550 --> 00:13:01.518 So we write down a over X minus one plus B over 00:13:01.518 --> 00:13:04.174 X minus 1 squared. 00:13:04.690 --> 00:13:08.700 Plus C over X +2. 00:13:10.490 --> 00:13:16.814 Again, we multiply through by the bottom line here, so we get 00:13:16.814 --> 00:13:23.665 a over X minus one times X minus one squared times X +2. 00:13:23.665 --> 00:13:30.516 One of the X minus ones will cancel, leaving us with 3X plus 00:13:30.516 --> 00:13:35.786 one equals a Times X minus one times X +2. 00:13:36.690 --> 00:13:43.256 B over X minus one squared times X minus one squared times X +2. 00:13:43.256 --> 00:13:49.353 Both of the X minus one squared will cancel, leaving us with B 00:13:49.353 --> 00:13:50.760 Times X +2. 00:13:51.700 --> 00:13:57.454 And then we have C over X +2 times X minus one squared times 00:13:57.454 --> 00:14:03.619 X +2. This time the X +2 is will cancel, leaving us with C Times 00:14:03.619 --> 00:14:05.263 X minus 1 squared. 00:14:07.090 --> 00:14:12.326 Again, the special values X equals one will make this zero, 00:14:12.326 --> 00:14:18.514 so a will disappear and it will make this zero. So see will 00:14:18.514 --> 00:14:24.041 disappear. If X equals one, we have 3X Plus One is 4. 00:14:24.650 --> 00:14:27.050 That zero so that expression 00:14:27.050 --> 00:14:32.080 disappears. 1 + 2 is 3, so we have 3B. 00:14:33.600 --> 00:14:40.544 This is 0, so this disappears. So we have 4 equals 3B. Giving B 00:14:40.544 --> 00:14:42.528 equals 4 over 3. 00:14:44.310 --> 00:14:47.268 If X equals. 00:14:47.990 --> 00:14:49.200 Minus 2. 00:14:51.390 --> 00:14:56.766 We have minus 2 * 3 is minus 6 Plus One is minus 5. 00:14:57.350 --> 00:15:00.542 Equals this is 0, so this 00:15:00.542 --> 00:15:05.398 disappears. This is 0, so this disappears minus 2. 00:15:05.990 --> 00:15:13.172 Minus one is minus 3 squared is 9, so we have minus five is 00:15:13.172 --> 00:15:16.763 9C, which gives us C is minus 00:15:16.763 --> 00:15:18.690 5. Over 9. 00:15:20.860 --> 00:15:28.324 We now need to find a. I'm just going to write this 00:15:28.324 --> 00:15:30.190 expression out again. 00:15:30.210 --> 00:15:43.062 I've written 00:15:43.062 --> 00:15:51.454 the. Expression following, see out 00:15:51.454 --> 00:15:55.486 like that because in a minute I'm going to multiply it out. 00:15:56.480 --> 00:16:00.062 Unfortunately, there's no special value of X that will 00:16:00.062 --> 00:16:02.450 eliminate B&C. To give us A. 00:16:03.130 --> 00:16:07.400 We can use any special value. We could use X equals 0. This would 00:16:07.400 --> 00:16:11.365 give us an equation in AB&C since we already know be in. See 00:16:11.365 --> 00:16:12.890 this would give us a. 00:16:13.590 --> 00:16:16.983 But I'm going to use a different technique, one 00:16:16.983 --> 00:16:19.245 called equating coefficients, and to do 00:16:19.245 --> 00:16:22.638 that I've got to multiply this lot right out. 00:16:24.330 --> 00:16:27.520 So we get equals a. 00:16:28.570 --> 00:16:31.486 And we have an X Times X for X 00:16:31.486 --> 00:16:37.294 squared. We have a minus 1X plus 2X, so that gives us Plus X. 00:16:38.080 --> 00:16:42.076 And we have minus 1 * 2 which gives us minus 2. 00:16:42.810 --> 00:16:46.082 And then plus BX 00:16:46.082 --> 00:16:49.390 +2. Plus C. 00:16:49.980 --> 00:16:55.860 X times X is X squared. We have a minus X under minus six, so 00:16:55.860 --> 00:17:00.956 that's minus 2X and then minus one times minus one is plus one. 00:17:02.250 --> 00:17:07.593 I'm not going to collect up all the terms. For instance, we have 00:17:07.593 --> 00:17:13.758 an A Times X squared here and we have a C Times X squared here. 00:17:13.758 --> 00:17:17.046 So we have a plus C Times X 00:17:17.046 --> 00:17:24.410 squared altogether. We also have an A Times XAB Times X&A minus 00:17:24.410 --> 00:17:26.690 two C Times X. 00:17:28.000 --> 00:17:34.180 A+B minus two C Times X? 00:17:34.730 --> 00:17:37.526 And finally we have minus 2A. 00:17:38.590 --> 00:17:40.738 2B and C. 00:17:41.260 --> 00:17:49.040 So the constant becomes minus 2A plus 2B Plus C. 00:17:51.380 --> 00:17:55.066 Now. So we have 3X plus one 00:17:55.066 --> 00:17:56.730 equals. This line. 00:17:57.900 --> 00:18:00.042 But in this line, we have a 00:18:00.042 --> 00:18:03.265 Turman X squared. 3X plus one doesn't have 00:18:03.265 --> 00:18:04.533 anything in X squared. 00:18:05.650 --> 00:18:10.900 But this is an identity. It must be true for all values of X, and 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 00:18:16.850 --> 00:18:21.400 that X squared disappears on this side. So we can say that a 00:18:21.400 --> 00:18:22.800 plus C equals 0. 00:18:23.750 --> 00:18:28.870 We already know that C is minus 5 over 9, so in order for 8 plus 00:18:28.870 --> 00:18:30.150 C to be 0. 00:18:30.720 --> 00:18:33.324 A must be plus five over 9. 00:18:33.870 --> 00:18:37.420 And we already worked out B as being equal to. 00:18:37.940 --> 00:18:42.802 For over 3, this means that we can write out the solution to 00:18:42.802 --> 00:18:43.924 the whole problem. 00:18:44.480 --> 00:18:51.704 3X plus one over X minus one squared 00:18:51.704 --> 00:18:54.413 times X +2. 00:18:55.000 --> 00:19:02.346 Equals. A5 over 9X minus one plus B is 00:19:02.346 --> 00:19:09.024 4 over 3 four over 3X minus 1 squared. 00:19:09.760 --> 00:19:16.774 See is minus 5 over 9, so we have minus five over 9 X 00:19:16.774 --> 00:19:23.000 +2. Another case we must consider. 00:19:23.540 --> 00:19:27.986 Is where the denominator contains a quadratic that can't 00:19:27.986 --> 00:19:30.950 be factorized as in 5X over. 00:19:31.460 --> 00:19:37.800 X squared plus X Plus One Times X minus 2. 00:19:38.500 --> 00:19:42.031 If we to express this in partial fractions, the two denominators 00:19:42.031 --> 00:19:46.525 are going to be X squared plus X Plus One and X minus 2. 00:19:47.120 --> 00:19:52.076 When the denominator is X squared plus 6 plus one, we have 00:19:52.076 --> 00:19:56.206 to consider the possibility that the numerator can contain a 00:19:56.206 --> 00:19:59.923 termine ex, because the numerator would still be of 00:19:59.923 --> 00:20:03.227 lower degree than the denominator, and this would 00:20:03.227 --> 00:20:08.596 still therefore be a proper fraction. So we write a X plus B 00:20:08.596 --> 00:20:11.487 over X squared plus X plus one. 00:20:12.090 --> 00:20:19.158 Plus C over X minus two as before. We multiply this out 00:20:19.158 --> 00:20:26.226 so we get that five X equals X plus B Times X 00:20:26.226 --> 00:20:33.454 minus 2. Plus C Times 00:20:33.454 --> 00:20:40.010 X squared. Plus 6 + 1. 00:20:41.630 --> 00:20:43.919 One special value we can use is 00:20:43.919 --> 00:20:46.510 X equals 2. And if. 00:20:47.020 --> 00:20:50.744 X equals 2, we 00:20:50.744 --> 00:20:57.005 get 5X5210. This is 0, so this all disappears and we 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. 00:21:04.940 --> 00:21:08.516 Giving C equals 10 over 7. 00:21:09.520 --> 00:21:14.200 Unfortunately, there's no value for X would enable us to get rid 00:21:14.200 --> 00:21:19.270 of C, so we're going to have to use the technique of equating 00:21:19.270 --> 00:21:25.770 coefficients. I'll write this out again. 00:21:26.450 --> 00:21:32.430 In order to equate 00:21:32.430 --> 00:21:38.410 coefficients, I'm going to 00:21:38.410 --> 00:21:44.390 have to multiply this 00:21:44.390 --> 00:21:45.885 out. 00:21:46.900 --> 00:21:50.026 X times X is X squared. 00:21:51.140 --> 00:21:54.794 X times minus two is minus two 00:21:54.794 --> 00:22:01.640 AX. B times X is BXB times minus two 00:22:01.640 --> 00:22:08.740 gives us minus 2B Plus CX squared Plus CX Plus 00:22:08.740 --> 00:22:13.806 C. Again, I'm going to collect like terms. So for instance for 00:22:13.806 --> 00:22:15.190 X squared we have. 00:22:16.380 --> 00:22:23.090 AX squared and CX squared. So we have a plus 00:22:23.090 --> 00:22:29.800 CX squared for X. We have a minus two AAB&C. 00:22:30.310 --> 00:22:38.200 So minus two A+B Plus CX and for a constant 00:22:38.200 --> 00:22:42.145 we have minus 2B Plus 00:22:42.145 --> 00:22:48.475 C. We still need to find 00:22:48.475 --> 00:22:50.609 both A&B. 00:22:51.870 --> 00:22:55.734 For two unknowns we need 2 equations, so we are going to 00:22:55.734 --> 00:22:57.666 have to solve for two different 00:22:57.666 --> 00:23:02.438 coefficients. Now the left hand side is just 5X, so there is no 00:23:02.438 --> 00:23:03.674 coefficient in X squared. 00:23:04.280 --> 00:23:09.936 In order to eliminate X squared, we can say that a plus C equals 00:23:09.936 --> 00:23:17.465 0. We already know what see is 10 over 7. In order for a plus C 00:23:17.465 --> 00:23:22.008 to be 0, this will make a minus 10 over 7. 00:23:24.280 --> 00:23:28.700 The left hand side also has no constant coefficient, so 00:23:28.700 --> 00:23:33.120 that means that this expression must be 0. So we 00:23:33.120 --> 00:23:36.214 say minus 2B Plus C equals 0. 00:23:37.620 --> 00:23:40.876 Giving us. C equals 00:23:40.876 --> 00:23:48.100 2B. Or B equals C over two, which gives us B as being. 00:23:48.610 --> 00:23:51.079 5 over 7. 00:23:52.960 --> 00:24:00.352 So we have a equal to minus 10 over 7B equal to 00:24:00.352 --> 00:24:06.512 five over 7 and C equal to 10 over 7. 00:24:08.000 --> 00:24:11.200 This means that 5X over. 00:24:11.700 --> 00:24:15.126 X squared plus X plus one. 00:24:15.670 --> 00:24:23.090 Times X minus two is equal to a X which is minus 10 over 00:24:23.090 --> 00:24:30.932 7X. Plus B, which is 5 over 7 all over X squared plus 00:24:30.932 --> 00:24:32.585 X plus one. 00:24:33.780 --> 00:24:40.892 Plus C, which is 10 over 7 over X minus two and are now 00:24:40.892 --> 00:24:47.496 tidy. This up the Seven comes down to be multiplied by the X 00:24:47.496 --> 00:24:54.608 squared plus X plus one. So we get minus 10X plus five over 7 00:24:54.608 --> 00:25:01.212 X squared plus X Plus One plus and again the Seven comes down 00:25:01.212 --> 00:25:03.752 10 over 7X minus 2. 00:25:03.860 --> 00:25:10.202 Equals and to finish it off we need to take five out of this 00:25:10.202 --> 00:25:15.185 expression as a factor, which gives us five times minus 2X 00:25:15.185 --> 00:25:22.433 plus one over 7 X squared plus X plus 1 + 10 over 7X minus 2. 00:25:23.300 --> 00:25:29.160 So far I've only dealt with proper fractions where the 00:25:29.160 --> 00:25:35.020 numerator is of lower degree than the denominator. Now, like 00:25:35.020 --> 00:25:38.536 to look at an improper fraction. 00:25:39.220 --> 00:25:40.950 Let's Express. 00:25:42.170 --> 00:25:49.202 4X cubed plus 10X plus four over X 00:25:49.202 --> 00:25:52.718 into 2X plus one. 00:25:53.390 --> 00:25:55.358 In partial fractions. 00:25:57.450 --> 00:25:59.460 The numerator is of degree 3. 00:26:00.590 --> 00:26:05.105 The denominator, if you multiply the X by the two X, you get 2 X 00:26:05.105 --> 00:26:06.610 squared, so the denominator is 00:26:06.610 --> 00:26:10.576 of degree 2. This means that this is an improper fraction. 00:26:11.750 --> 00:26:15.998 What this means is that if you divide the numerator by the 00:26:15.998 --> 00:26:19.892 denominator, you're going to be dividing otermin X cubed by a 00:26:19.892 --> 00:26:21.308 term in X squared. 00:26:22.180 --> 00:26:24.640 So you could get a Terminix. 00:26:25.160 --> 00:26:29.788 Which means that we have to write down acts. We may also get 00:26:29.788 --> 00:26:33.348 a constant term, so we have to write down B. 00:26:33.910 --> 00:26:36.280 Then we can do our fractions. 00:26:37.130 --> 00:26:44.510 If I now multiply but through I get a X 00:26:44.510 --> 00:26:51.890 Times X Times 2X plus one, so we get 4X 00:26:51.890 --> 00:26:55.580 cubed plus 10X plus four 00:26:55.580 --> 00:26:58.850 equals a. X squared 00:26:59.410 --> 00:27:02.558 Times 2X plus one. 00:27:02.560 --> 00:27:06.250 Plus BX times 2X 00:27:06.250 --> 00:27:13.206 plus one. Plus C Times 2X plus one. 00:27:14.500 --> 00:27:17.670 Plus DX 00:27:21.530 --> 00:27:23.039 Using special values. 00:27:23.730 --> 00:27:29.456 If I use X equals 0, then the term the D, the B, and 00:27:29.456 --> 00:27:33.955 the A are all going to disappear and I'm just left 00:27:33.955 --> 00:27:36.818 with see. So if X equals 0. 00:27:37.930 --> 00:27:42.675 X cubed is zero, X is zero. I just get 4 equal to. 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 00:27:50.706 --> 00:27:53.082 special value is X equal to 00:27:53.082 --> 00:27:59.391 minus 1/2. If X equals minus Alpha, this is 0, so this will 00:27:59.391 --> 00:28:02.397 disappear. This is 0, so this 00:28:02.397 --> 00:28:06.696 will disappear. And this will disappear, just leaving me with 00:28:06.696 --> 00:28:09.320 D. So I get. 00:28:10.330 --> 00:28:15.263 Minus 1/2. Cubed is minus an eighth, so we 00:28:15.263 --> 00:28:17.508 get minus four over 8. 00:28:18.650 --> 00:28:25.922 Plus 10 times minus 1/2 inches minus 10 over 2 + 4 00:28:25.922 --> 00:28:29.380 equals. D times minus 00:28:29.380 --> 00:28:36.478 1/2. I'll just write that down again, 00:28:36.478 --> 00:28:44.182 minus four over 8 - 10 over 2. 00:28:45.480 --> 00:28:49.340 +4. Equals minus 00:28:49.340 --> 00:28:54.704 1/2 D. Minus 4 over 8 is just minus 1/2. 00:28:55.290 --> 00:29:02.440 Minus 10 over 2 is minus 5 + 4 equals minus half D. 00:29:04.290 --> 00:29:11.584 Minus 5 + 4 is minus one, so I've got minus 1 1/2 equals 00:29:11.584 --> 00:29:13.147 minus 1/2 D. 00:29:16.230 --> 00:29:21.433 Minus 1 1/2 is just three times minus 1/2, so this 00:29:21.433 --> 00:29:23.798 gives us D equal 3. 00:29:25.090 --> 00:29:31.054 Special values won't give me a or be, so I'm going to have to 00:29:31.054 --> 00:29:35.314 equate coefficients. This means I have to write this expression 00:29:35.314 --> 00:29:40.875 out again. 4X cubed plus 00:29:40.875 --> 00:29:48.198 10X. +4 equals a X squared times. 00:29:48.770 --> 00:29:55.218 2X plus one plus BX times 2X plus 00:29:55.218 --> 00:29:57.230 one. Plus 00:29:58.260 --> 00:30:03.860 C times 2X plus one plus DX. 00:30:05.550 --> 00:30:07.496 I'm now going to multiply this out. 00:30:08.940 --> 00:30:13.860 X squared times 2X is 2A X cubed. 00:30:14.920 --> 00:30:18.864 X squared times one is just X squared. 00:30:19.960 --> 00:30:22.438 This gives me 2B X squared. 00:30:24.850 --> 00:30:27.961 This gives me 00:30:27.961 --> 00:30:31.576 BX. This gives Me 2 00:30:31.576 --> 00:30:36.818 CX. This gives me C. 00:30:37.990 --> 00:30:38.660 And then. 00:30:39.840 --> 00:30:47.330 Plus DX And collecting terms, we only have one Turman X cubed, so 00:30:47.330 --> 00:30:50.330 that is just 2A X cubed. 00:30:51.010 --> 00:30:55.993 Plus we have two terms in X squared, A and 2B. 00:30:56.830 --> 00:31:04.550 We have three terms in XB2C and D. 00:31:05.120 --> 00:31:10.632 And finally, the constant 00:31:10.632 --> 00:31:13.388 term see. 00:31:15.050 --> 00:31:19.572 Now look at the Turman X cubed. We have 4X cubed on the left. 00:31:20.330 --> 00:31:25.860 And two AX cubed on the right. This means that 2A must be equal 00:31:25.860 --> 00:31:32.282 to 4. Giving us a equal to two now look at the Turman X 00:31:32.282 --> 00:31:35.850 squared. There is no Turman X squared on the left. 00:31:36.600 --> 00:31:39.228 And on the right we have a plus 2B. 00:31:40.310 --> 00:31:43.729 This means that as there isn't Turman X squared on the left, a 00:31:43.729 --> 00:31:45.570 plus 2B must be equal to 0. 00:31:46.120 --> 00:31:49.648 So we have a plus 2B 00:31:49.648 --> 00:31:52.820 equals 0. Which means that. 00:31:53.450 --> 00:31:56.951 A equals minus 00:31:56.951 --> 00:32:01.560 2B. Which means that B equals. 00:32:03.040 --> 00:32:07.779 Minus two over 2 equals minus one. 00:32:09.110 --> 00:32:15.088 I'll just write those values in again. 00:32:15.800 --> 00:32:17.768 A equals 2. 00:32:18.510 --> 00:32:24.936 B equals minus one C equals 4D equals 3. 00:32:27.700 --> 00:32:34.124 So if we take our original expression 4X cubed plus 10X 00:32:34.124 --> 00:32:37.044 plus four over X times. 00:32:38.890 --> 00:32:44.810 2X plus one. This is equal to axe, so 2X. 00:32:45.390 --> 00:32:52.431 Minus B. Plus see over X, so that's four over X Plus D 00:32:52.431 --> 00:32:57.282 over 2X Plus One which is 3 over 2X plus one.