[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:01.36,0:00:05.75,Default,,0000,0000,0000,,An algebraic fraction is one\Nwhere the numerator and Dialogue: 0,0:00:05.75,0:00:07.70,Default,,0000,0000,0000,,denominator, both polynomial\Nexpressions. Dialogue: 0,0:00:08.86,0:00:12.88,Default,,0000,0000,0000,,This is an expression where\Nevery term is a multiple of a Dialogue: 0,0:00:12.88,0:00:14.22,Default,,0000,0000,0000,,power of X, like. Dialogue: 0,0:00:14.83,0:00:21.42,Default,,0000,0000,0000,,5X to the 4th\Nplus 6X cubed plus Dialogue: 0,0:00:21.42,0:00:23.89,Default,,0000,0000,0000,,7X plus 4. Dialogue: 0,0:00:24.86,0:00:28.81,Default,,0000,0000,0000,,The degree of a polynomial is\Nthe power of the highest Dialogue: 0,0:00:28.81,0:00:32.04,Default,,0000,0000,0000,,Terminix. So this is a\Npolynomial of degree 4. Dialogue: 0,0:00:33.40,0:00:37.95,Default,,0000,0000,0000,,The number in front of X in each\Ncase is the coefficient of that Dialogue: 0,0:00:37.95,0:00:42.50,Default,,0000,0000,0000,,term. So the coefficient of X to\Nthe 4th is 5. The coefficient of Dialogue: 0,0:00:42.50,0:00:43.80,Default,,0000,0000,0000,,X cubed is 6. Dialogue: 0,0:00:45.12,0:00:52.70,Default,,0000,0000,0000,,Now look at these fractions\NX over X squared +2. Dialogue: 0,0:00:53.35,0:00:59.59,Default,,0000,0000,0000,,Or X cubed plus three\Nover X to the 4th plus Dialogue: 0,0:00:59.59,0:01:01.86,Default,,0000,0000,0000,,X squared plus one. Dialogue: 0,0:01:02.90,0:01:04.35,Default,,0000,0000,0000,,In both cases. Dialogue: 0,0:01:04.98,0:01:07.21,Default,,0000,0000,0000,,The numerator is a polynomial of Dialogue: 0,0:01:07.21,0:01:09.38,Default,,0000,0000,0000,,lower degree. Then the Dialogue: 0,0:01:09.38,0:01:16.05,Default,,0000,0000,0000,,denominator. X's against X\Nsquared X cubed as against X to Dialogue: 0,0:01:16.05,0:01:19.30,Default,,0000,0000,0000,,the 4th week. All these proper Dialogue: 0,0:01:19.30,0:01:24.89,Default,,0000,0000,0000,,fractions. With other fractions,\Nthe polynomial may be of higher Dialogue: 0,0:01:24.89,0:01:30.49,Default,,0000,0000,0000,,degree in the numerator. For\Ninstance, X fourth plus X Dialogue: 0,0:01:30.49,0:01:32.17,Default,,0000,0000,0000,,squared plus X. Dialogue: 0,0:01:33.17,0:01:41.01,Default,,0000,0000,0000,,Over X cubed plus X +2 or\Nit may be of the same degree. Dialogue: 0,0:01:41.68,0:01:49.08,Default,,0000,0000,0000,,Such as X plus four over\NX plus three. We call these Dialogue: 0,0:01:49.08,0:01:55.49,Default,,0000,0000,0000,,improper fractions. Down,\Nlook like to look at Dialogue: 0,0:01:55.49,0:02:02.47,Default,,0000,0000,0000,,how we add and subtract\Nfractions. Take for instance Dialogue: 0,0:02:02.47,0:02:04.80,Default,,0000,0000,0000,,these two fractions. Dialogue: 0,0:02:06.87,0:02:14.16,Default,,0000,0000,0000,,In order to add these two\Nfractions together, we need to Dialogue: 0,0:02:14.16,0:02:19.47,Default,,0000,0000,0000,,find the lowest common\Ndenominator. In this particular Dialogue: 0,0:02:19.47,0:02:27.42,Default,,0000,0000,0000,,case it's X minus 3 *\N2 X plus one, so we Dialogue: 0,0:02:27.42,0:02:31.40,Default,,0000,0000,0000,,say that this sum is the Dialogue: 0,0:02:31.40,0:02:36.11,Default,,0000,0000,0000,,equivalent of. In the\Ndenominators we are going to Dialogue: 0,0:02:36.11,0:02:39.33,Default,,0000,0000,0000,,have X minus 3 * 2 X plus one. Dialogue: 0,0:02:42.20,0:02:49.35,Default,,0000,0000,0000,,In order to get from there to\Nthere, we multiplied by 2X plus Dialogue: 0,0:02:49.35,0:02:54.85,Default,,0000,0000,0000,,one, we've multiplied the\Ndenominator by 2X plus one. So Dialogue: 0,0:02:54.85,0:02:59.80,Default,,0000,0000,0000,,we must multiply the numerator\Nby 2X plus one. Dialogue: 0,0:02:59.88,0:03:05.53,Default,,0000,0000,0000,,And in order to get from here to\Nhere, we've multiplied the Dialogue: 0,0:03:05.53,0:03:10.71,Default,,0000,0000,0000,,denominator by X minus three. So\Nwe've got to multiply the Dialogue: 0,0:03:10.71,0:03:16.36,Default,,0000,0000,0000,,numerator by X minus three, and\Nthis gives us just X minus Dialogue: 0,0:03:16.36,0:03:18.36,Default,,0000,0000,0000,,three. Now we need to collect Dialogue: 0,0:03:18.36,0:03:23.31,Default,,0000,0000,0000,,up. The denominators of the\Nsame, so we can just right. Dialogue: 0,0:03:23.95,0:03:31.34,Default,,0000,0000,0000,,X minus 3 * 2 X Plus\NOne and on top we have 2 Dialogue: 0,0:03:31.34,0:03:36.62,Default,,0000,0000,0000,,* 2 X is 4X Minus, X\Ngives us 3X. Dialogue: 0,0:03:38.26,0:03:41.17,Default,,0000,0000,0000,,And we also have 2 * 1 is 2. Dialogue: 0,0:03:42.32,0:03:49.25,Default,,0000,0000,0000,,Minus minus three is plus three,\Nso 2 + 3 is +5 and that Dialogue: 0,0:03:49.25,0:03:52.22,Default,,0000,0000,0000,,is the answer to that some. Dialogue: 0,0:03:53.01,0:04:00.20,Default,,0000,0000,0000,,Sometimes in mathematics we need\Nto do this operation in reverse. Dialogue: 0,0:04:00.20,0:04:06.74,Default,,0000,0000,0000,,In calculus, for instance, or\Nwhen dealing with the binomial Dialogue: 0,0:04:06.74,0:04:11.56,Default,,0000,0000,0000,,theorem. We sometimes need to\Nsplit a fraction up into its Dialogue: 0,0:04:11.56,0:04:14.20,Default,,0000,0000,0000,,component parts, which are\Ncalled partial fractions. Let's Dialogue: 0,0:04:14.20,0:04:18.16,Default,,0000,0000,0000,,take the sum that I've just\Ndealt with. We got the answer. Dialogue: 0,0:04:18.70,0:04:21.28,Default,,0000,0000,0000,,Three X +5. Dialogue: 0,0:04:21.80,0:04:22.49,Default,,0000,0000,0000,,Over. Dialogue: 0,0:04:24.41,0:04:29.98,Default,,0000,0000,0000,,X minus 3 * 2\NX plus one. Dialogue: 0,0:04:30.59,0:04:35.20,Default,,0000,0000,0000,,So how do we get this back to\Nits component parts? Well? Dialogue: 0,0:04:36.12,0:04:40.56,Default,,0000,0000,0000,,We only have two factors in the\Ndenominator, X minus three and Dialogue: 0,0:04:40.56,0:04:41.67,Default,,0000,0000,0000,,2X plus one. Dialogue: 0,0:04:42.87,0:04:49.51,Default,,0000,0000,0000,,So. It must be something\Nover X minus three plus Dialogue: 0,0:04:49.51,0:04:52.84,Default,,0000,0000,0000,,something. Over 2X plus one. Dialogue: 0,0:04:53.90,0:04:59.14,Default,,0000,0000,0000,,And what are these some things?\NThey can only be plain numbers, Dialogue: 0,0:04:59.14,0:05:04.82,Default,,0000,0000,0000,,because if they involved X or\Npowers of X then these would be Dialogue: 0,0:05:04.82,0:05:09.20,Default,,0000,0000,0000,,improper fractions, so we're\Nquite entitled to say that 3X Dialogue: 0,0:05:09.20,0:05:16.62,Default,,0000,0000,0000,,plus five over X minus 3 * 2 X\NPlus one is a over X minus three Dialogue: 0,0:05:16.62,0:05:21.87,Default,,0000,0000,0000,,plus B over 2X plus one where\NA&B are just plain numbers. Dialogue: 0,0:05:22.80,0:05:26.94,Default,,0000,0000,0000,,The next thing to do is to\Nmultiply everything through by Dialogue: 0,0:05:26.94,0:05:31.45,Default,,0000,0000,0000,,what's on the bottom X minus 3 *\N2 X plus one. Dialogue: 0,0:05:32.05,0:05:37.04,Default,,0000,0000,0000,,If you multiply the left hand\Nside by that, we just get three Dialogue: 0,0:05:37.04,0:05:38.19,Default,,0000,0000,0000,,X +5 equals. Dialogue: 0,0:05:39.24,0:05:46.32,Default,,0000,0000,0000,,A over X minus three times X\Nminus 3 * 2 X plus one the Dialogue: 0,0:05:46.32,0:05:51.51,Default,,0000,0000,0000,,X minus threes will cancel,\Nand we're just left with a Dialogue: 0,0:05:51.51,0:05:53.40,Default,,0000,0000,0000,,Times 2X plus one. Dialogue: 0,0:05:55.72,0:06:02.71,Default,,0000,0000,0000,,B over 2X Plus One Times X minus\N3 * 2 X Plus one. This time the Dialogue: 0,0:06:02.71,0:06:08.46,Default,,0000,0000,0000,,2X plus ones will cancel and we\Njust left with B Times X minus Dialogue: 0,0:06:08.46,0:06:14.75,Default,,0000,0000,0000,,three. Now this is an identity,\Nwhich means that it is true for Dialogue: 0,0:06:14.75,0:06:16.52,Default,,0000,0000,0000,,all values of X. Dialogue: 0,0:06:17.34,0:06:20.74,Default,,0000,0000,0000,,If this is so, then we can\Nsubstitute special values for X Dialogue: 0,0:06:20.74,0:06:22.43,Default,,0000,0000,0000,,and it will still be true. Dialogue: 0,0:06:23.47,0:06:27.16,Default,,0000,0000,0000,,For instance, if we make X equal\Nto minus 1/2. Dialogue: 0,0:06:28.33,0:06:31.36,Default,,0000,0000,0000,,This bracket will become zero\Nand a will disappear. Dialogue: 0,0:06:32.44,0:06:36.27,Default,,0000,0000,0000,,If we make X equal to three,\Nthis bracket will become zero Dialogue: 0,0:06:36.27,0:06:37.54,Default,,0000,0000,0000,,and be will disappear. Dialogue: 0,0:06:38.26,0:06:39.88,Default,,0000,0000,0000,,And I'm going to do just that. Dialogue: 0,0:06:40.59,0:06:43.91,Default,,0000,0000,0000,,If X equals minus 1/2. Dialogue: 0,0:06:44.52,0:06:51.49,Default,,0000,0000,0000,,We get three times minus 1/2\Nis minus three over 2 + Dialogue: 0,0:06:51.49,0:06:54.13,Default,,0000,0000,0000,,5. That is 0. Dialogue: 0,0:06:55.20,0:07:02.09,Default,,0000,0000,0000,,Equals B times minus\N1/2 - 3. Dialogue: 0,0:07:03.04,0:07:09.87,Default,,0000,0000,0000,,This is just Seven over\N2 and we get 7 over 2 Dialogue: 0,0:07:09.87,0:07:16.70,Default,,0000,0000,0000,,equals. This is minus 7\Nover 2 - 7 over 2B so Dialogue: 0,0:07:16.70,0:07:20.11,Default,,0000,0000,0000,,B is equal to minus one. Dialogue: 0,0:07:22.99,0:07:29.74,Default,,0000,0000,0000,,All right, this line\Nin again 3X plus Dialogue: 0,0:07:29.74,0:07:35.67,Default,,0000,0000,0000,,5. Equals a Times\N2X plus one. Dialogue: 0,0:07:36.28,0:07:38.77,Default,,0000,0000,0000,,Plus B times. Dialogue: 0,0:07:40.52,0:07:42.75,Default,,0000,0000,0000,,X minus three. Dialogue: 0,0:07:43.33,0:07:47.31,Default,,0000,0000,0000,,This time I want to try and find\Na, so I'm going to put X equal Dialogue: 0,0:07:47.31,0:07:50.71,Default,,0000,0000,0000,,to 3. If X equals Dialogue: 0,0:07:50.71,0:07:56.86,Default,,0000,0000,0000,,3. We have 3 threes and\N9 + 5 is 14. Dialogue: 0,0:07:57.79,0:08:00.38,Default,,0000,0000,0000,,3266 plus One is 7. Dialogue: 0,0:08:02.19,0:08:07.74,Default,,0000,0000,0000,,That is going to be 0, so be\Nwill disappear, so A is equal to Dialogue: 0,0:08:07.74,0:08:10.70,Default,,0000,0000,0000,,14 / 7. In other words, a IS2. Dialogue: 0,0:08:11.94,0:08:15.56,Default,,0000,0000,0000,,We already had the\Nequal to minus one. Dialogue: 0,0:08:16.82,0:08:19.06,Default,,0000,0000,0000,,So what do we have now? Dialogue: 0,0:08:20.59,0:08:24.38,Default,,0000,0000,0000,,I had three X +5 over. Dialogue: 0,0:08:25.51,0:08:27.75,Default,,0000,0000,0000,,X minus three. Dialogue: 0,0:08:28.64,0:08:34.54,Default,,0000,0000,0000,,2X plus one times 2X plus\None equals a over. Dialogue: 0,0:08:35.14,0:08:42.74,Default,,0000,0000,0000,,X minus three plus B over 2X\Nplus one. Since A is 2 and Dialogue: 0,0:08:42.74,0:08:50.34,Default,,0000,0000,0000,,B is minus one, we can see\Nthat this is 2 over X minus Dialogue: 0,0:08:50.34,0:08:52.52,Default,,0000,0000,0000,,three plus, sorry minus. Dialogue: 0,0:08:53.41,0:08:58.54,Default,,0000,0000,0000,,One over 2X Plus One, which is\Nthe sum that we started with Dialogue: 0,0:08:58.54,0:09:02.89,Default,,0000,0000,0000,,and we have now broken this\Nback into its component parts Dialogue: 0,0:09:02.89,0:09:04.08,Default,,0000,0000,0000,,called partial fractions. Dialogue: 0,0:09:05.29,0:09:12.71,Default,,0000,0000,0000,,Do another example. Let's say\Nthat we have to express Dialogue: 0,0:09:12.71,0:09:20.13,Default,,0000,0000,0000,,3X over X minus one\Ntimes X +2 in partial Dialogue: 0,0:09:20.13,0:09:26.44,Default,,0000,0000,0000,,fractions. Again, we look at the\Ndenominator. The factors in the Dialogue: 0,0:09:26.44,0:09:32.21,Default,,0000,0000,0000,,denominator X minus one and X\N+2. So we say that this Dialogue: 0,0:09:32.21,0:09:38.95,Default,,0000,0000,0000,,expression is equal to a over X\Nminus one plus B over X +2. Dialogue: 0,0:09:40.48,0:09:45.100,Default,,0000,0000,0000,,We multiply through by X minus\None times X +2 on the left hand Dialogue: 0,0:09:45.100,0:09:48.36,Default,,0000,0000,0000,,side. This just gives us 3X. Dialogue: 0,0:09:50.44,0:09:56.56,Default,,0000,0000,0000,,On the right hand side, a over X\Nminus one times X minus one Dialogue: 0,0:09:56.56,0:10:02.68,Default,,0000,0000,0000,,times X +2 X minus ones cancel\Nout, and we're left with a Times Dialogue: 0,0:10:02.68,0:10:09.90,Default,,0000,0000,0000,,X +2. Be over X +2\Ntimes X minus 1X Plus 2X Dialogue: 0,0:10:09.90,0:10:16.49,Default,,0000,0000,0000,,plus Two's cancel out and\Nwe're left with B Times X Dialogue: 0,0:10:16.49,0:10:17.69,Default,,0000,0000,0000,,minus one. Dialogue: 0,0:10:21.08,0:10:25.16,Default,,0000,0000,0000,,This time the special values\Nthat I'm going to take our X Dialogue: 0,0:10:25.16,0:10:28.90,Default,,0000,0000,0000,,equals minus two because that\Nwill make that zero and thus Dialogue: 0,0:10:28.90,0:10:32.98,Default,,0000,0000,0000,,eliminate A and X equals 1,\Nwhich will make that zero and Dialogue: 0,0:10:32.98,0:10:34.00,Default,,0000,0000,0000,,thus eliminate B. Dialogue: 0,0:10:35.03,0:10:38.25,Default,,0000,0000,0000,,If X equals minus Dialogue: 0,0:10:38.25,0:10:44.83,Default,,0000,0000,0000,,2. We get three times\Nminus two is minus 6. Dialogue: 0,0:10:45.63,0:10:48.04,Default,,0000,0000,0000,,That is 0, so a disappears. Dialogue: 0,0:10:49.12,0:10:55.44,Default,,0000,0000,0000,,Minus 2 - 1 is minus three,\Nso this is minus 3B. Dialogue: 0,0:10:56.07,0:10:56.79,Default,,0000,0000,0000,,So. Dialogue: 0,0:10:57.81,0:11:04.95,Default,,0000,0000,0000,,B equals minus 6 divided\Nby minus 3 equals 2. Dialogue: 0,0:11:06.13,0:11:12.79,Default,,0000,0000,0000,,Alright, this\Nexpression in Dialogue: 0,0:11:12.79,0:11:14.46,Default,,0000,0000,0000,,again. Dialogue: 0,0:11:16.11,0:11:23.46,Default,,0000,0000,0000,,This time I'm\Ngoing to put Dialogue: 0,0:11:23.46,0:11:27.14,Default,,0000,0000,0000,,X equal to Dialogue: 0,0:11:27.14,0:11:30.69,Default,,0000,0000,0000,,1. 3 * 1 Dialogue: 0,0:11:30.69,0:11:37.74,Default,,0000,0000,0000,,is 3. 1 + 2\Nis 3, so we get 3A. Dialogue: 0,0:11:38.35,0:11:41.41,Default,,0000,0000,0000,,1 - 1 is 0 so be disappears. Dialogue: 0,0:11:42.55,0:11:48.18,Default,,0000,0000,0000,,If 3A equals 3, then a is\Ngoing to equal 1, so we've Dialogue: 0,0:11:48.18,0:11:52.94,Default,,0000,0000,0000,,got a equal 1. We already\Nhad B equal to two. Dialogue: 0,0:11:54.38,0:11:56.15,Default,,0000,0000,0000,,I'm not going to write the whole Dialogue: 0,0:11:56.15,0:11:59.36,Default,,0000,0000,0000,,expression in again. We have 3X. Dialogue: 0,0:11:59.87,0:12:07.55,Default,,0000,0000,0000,,Over X minus one\Ntimes X +2 equals. Dialogue: 0,0:12:08.05,0:12:15.50,Default,,0000,0000,0000,,One over X minus one because a\Nis 1 + 2 over X +2 because Dialogue: 0,0:12:15.50,0:12:19.48,Default,,0000,0000,0000,,be is 2 and that is the answer. Dialogue: 0,0:12:20.29,0:12:28.15,Default,,0000,0000,0000,,Sometimes the denominators more\Nawkward, for example, to Dialogue: 0,0:12:28.15,0:12:36.02,Default,,0000,0000,0000,,express 3X plus one\Nover X minus one Dialogue: 0,0:12:36.02,0:12:39.95,Default,,0000,0000,0000,,squared times X +2. Dialogue: 0,0:12:40.90,0:12:43.90,Default,,0000,0000,0000,,There are actually three\Npossibilities for a denominator Dialogue: 0,0:12:43.90,0:12:45.40,Default,,0000,0000,0000,,in the partial fraction. Dialogue: 0,0:12:45.93,0:12:50.91,Default,,0000,0000,0000,,We've got X minus One X +2, but\Nthere's also the possibility of Dialogue: 0,0:12:50.91,0:12:52.44,Default,,0000,0000,0000,,X minus 1 squared. Dialogue: 0,0:12:53.55,0:13:01.52,Default,,0000,0000,0000,,So we write down a over\NX minus one plus B over Dialogue: 0,0:13:01.52,0:13:04.17,Default,,0000,0000,0000,,X minus 1 squared. Dialogue: 0,0:13:04.69,0:13:08.70,Default,,0000,0000,0000,,Plus C over X\N+2. Dialogue: 0,0:13:10.49,0:13:16.81,Default,,0000,0000,0000,,Again, we multiply through by\Nthe bottom line here, so we get Dialogue: 0,0:13:16.81,0:13:23.66,Default,,0000,0000,0000,,a over X minus one times X\Nminus one squared times X +2. Dialogue: 0,0:13:23.66,0:13:30.52,Default,,0000,0000,0000,,One of the X minus ones will\Ncancel, leaving us with 3X plus Dialogue: 0,0:13:30.52,0:13:35.79,Default,,0000,0000,0000,,one equals a Times X minus one\Ntimes X +2. Dialogue: 0,0:13:36.69,0:13:43.26,Default,,0000,0000,0000,,B over X minus one squared times\NX minus one squared times X +2. Dialogue: 0,0:13:43.26,0:13:49.35,Default,,0000,0000,0000,,Both of the X minus one squared\Nwill cancel, leaving us with B Dialogue: 0,0:13:49.35,0:13:50.76,Default,,0000,0000,0000,,Times X +2. Dialogue: 0,0:13:51.70,0:13:57.45,Default,,0000,0000,0000,,And then we have C over X +2\Ntimes X minus one squared times Dialogue: 0,0:13:57.45,0:14:03.62,Default,,0000,0000,0000,,X +2. This time the X +2 is will\Ncancel, leaving us with C Times Dialogue: 0,0:14:03.62,0:14:05.26,Default,,0000,0000,0000,,X minus 1 squared. Dialogue: 0,0:14:07.09,0:14:12.33,Default,,0000,0000,0000,,Again, the special values X\Nequals one will make this zero, Dialogue: 0,0:14:12.33,0:14:18.51,Default,,0000,0000,0000,,so a will disappear and it will\Nmake this zero. So see will Dialogue: 0,0:14:18.51,0:14:24.04,Default,,0000,0000,0000,,disappear. If X equals one, we\Nhave 3X Plus One is 4. Dialogue: 0,0:14:24.65,0:14:27.05,Default,,0000,0000,0000,,That zero so that expression Dialogue: 0,0:14:27.05,0:14:32.08,Default,,0000,0000,0000,,disappears. 1 + 2 is 3, so\Nwe have 3B. Dialogue: 0,0:14:33.60,0:14:40.54,Default,,0000,0000,0000,,This is 0, so this disappears.\NSo we have 4 equals 3B. Giving B Dialogue: 0,0:14:40.54,0:14:42.53,Default,,0000,0000,0000,,equals 4 over 3. Dialogue: 0,0:14:44.31,0:14:47.27,Default,,0000,0000,0000,,If X equals. Dialogue: 0,0:14:47.99,0:14:49.20,Default,,0000,0000,0000,,Minus 2. Dialogue: 0,0:14:51.39,0:14:56.77,Default,,0000,0000,0000,,We have minus 2 * 3 is minus 6\NPlus One is minus 5. Dialogue: 0,0:14:57.35,0:15:00.54,Default,,0000,0000,0000,,Equals this is 0, so this Dialogue: 0,0:15:00.54,0:15:05.40,Default,,0000,0000,0000,,disappears. This is 0, so this\Ndisappears minus 2. Dialogue: 0,0:15:05.99,0:15:13.17,Default,,0000,0000,0000,,Minus one is minus 3 squared is\N9, so we have minus five is Dialogue: 0,0:15:13.17,0:15:16.76,Default,,0000,0000,0000,,9C, which gives us C is minus Dialogue: 0,0:15:16.76,0:15:18.69,Default,,0000,0000,0000,,5. Over 9. Dialogue: 0,0:15:20.86,0:15:28.32,Default,,0000,0000,0000,,We now need to find a.\NI'm just going to write this Dialogue: 0,0:15:28.32,0:15:30.19,Default,,0000,0000,0000,,expression out again. Dialogue: 0,0:15:30.21,0:15:43.06,Default,,0000,0000,0000,,I've\Nwritten Dialogue: 0,0:15:43.06,0:15:51.45,Default,,0000,0000,0000,,the.\NExpression following, see out Dialogue: 0,0:15:51.45,0:15:55.49,Default,,0000,0000,0000,,like that because in a minute\NI'm going to multiply it out. Dialogue: 0,0:15:56.48,0:16:00.06,Default,,0000,0000,0000,,Unfortunately, there's no\Nspecial value of X that will Dialogue: 0,0:16:00.06,0:16:02.45,Default,,0000,0000,0000,,eliminate B&C. To give us A. Dialogue: 0,0:16:03.13,0:16:07.40,Default,,0000,0000,0000,,We can use any special value. We\Ncould use X equals 0. This would Dialogue: 0,0:16:07.40,0:16:11.36,Default,,0000,0000,0000,,give us an equation in AB&C\Nsince we already know be in. See Dialogue: 0,0:16:11.36,0:16:12.89,Default,,0000,0000,0000,,this would give us a. Dialogue: 0,0:16:13.59,0:16:16.98,Default,,0000,0000,0000,,But I'm going to use a\Ndifferent technique, one Dialogue: 0,0:16:16.98,0:16:19.24,Default,,0000,0000,0000,,called equating\Ncoefficients, and to do Dialogue: 0,0:16:19.24,0:16:22.64,Default,,0000,0000,0000,,that I've got to multiply\Nthis lot right out. Dialogue: 0,0:16:24.33,0:16:27.52,Default,,0000,0000,0000,,So we get equals a. Dialogue: 0,0:16:28.57,0:16:31.49,Default,,0000,0000,0000,,And we have an X Times X for X Dialogue: 0,0:16:31.49,0:16:37.29,Default,,0000,0000,0000,,squared. We have a minus 1X plus\N2X, so that gives us Plus X. Dialogue: 0,0:16:38.08,0:16:42.08,Default,,0000,0000,0000,,And we have minus 1 * 2 which\Ngives us minus 2. Dialogue: 0,0:16:42.81,0:16:46.08,Default,,0000,0000,0000,,And then plus BX Dialogue: 0,0:16:46.08,0:16:49.39,Default,,0000,0000,0000,,+2. Plus C. Dialogue: 0,0:16:49.98,0:16:55.86,Default,,0000,0000,0000,,X times X is X squared. We have\Na minus X under minus six, so Dialogue: 0,0:16:55.86,0:17:00.96,Default,,0000,0000,0000,,that's minus 2X and then minus\None times minus one is plus one. Dialogue: 0,0:17:02.25,0:17:07.59,Default,,0000,0000,0000,,I'm not going to collect up all\Nthe terms. For instance, we have Dialogue: 0,0:17:07.59,0:17:13.76,Default,,0000,0000,0000,,an A Times X squared here and we\Nhave a C Times X squared here. Dialogue: 0,0:17:13.76,0:17:17.05,Default,,0000,0000,0000,,So we have a plus C Times X Dialogue: 0,0:17:17.05,0:17:24.41,Default,,0000,0000,0000,,squared altogether. We also have\Nan A Times XAB Times X&A minus Dialogue: 0,0:17:24.41,0:17:26.69,Default,,0000,0000,0000,,two C Times X. Dialogue: 0,0:17:28.00,0:17:34.18,Default,,0000,0000,0000,,A+B minus two\NC Times X? Dialogue: 0,0:17:34.73,0:17:37.53,Default,,0000,0000,0000,,And finally we have minus 2A. Dialogue: 0,0:17:38.59,0:17:40.74,Default,,0000,0000,0000,,2B and C. Dialogue: 0,0:17:41.26,0:17:49.04,Default,,0000,0000,0000,,So the constant becomes minus\N2A plus 2B Plus C. Dialogue: 0,0:17:51.38,0:17:55.07,Default,,0000,0000,0000,,Now. So we have 3X plus one Dialogue: 0,0:17:55.07,0:17:56.73,Default,,0000,0000,0000,,equals. This line. Dialogue: 0,0:17:57.90,0:18:00.04,Default,,0000,0000,0000,,But in this line, we have a Dialogue: 0,0:18:00.04,0:18:03.26,Default,,0000,0000,0000,,Turman X squared. 3X\Nplus one doesn't have Dialogue: 0,0:18:03.26,0:18:04.53,Default,,0000,0000,0000,,anything in X squared. Dialogue: 0,0:18:05.65,0:18:10.90,Default,,0000,0000,0000,,But this is an identity. It must\Nbe true for all values of X, and Dialogue: 0,0:18:10.90,0:18:16.85,Default,,0000,0000,0000,,the only way that this can be\Ntrue is for A plus E to be 0 so Dialogue: 0,0:18:16.85,0:18:21.40,Default,,0000,0000,0000,,that X squared disappears on\Nthis side. So we can say that a Dialogue: 0,0:18:21.40,0:18:22.80,Default,,0000,0000,0000,,plus C equals 0. Dialogue: 0,0:18:23.75,0:18:28.87,Default,,0000,0000,0000,,We already know that C is minus\N5 over 9, so in order for 8 plus Dialogue: 0,0:18:28.87,0:18:30.15,Default,,0000,0000,0000,,C to be 0. Dialogue: 0,0:18:30.72,0:18:33.32,Default,,0000,0000,0000,,A must be plus five over 9. Dialogue: 0,0:18:33.87,0:18:37.42,Default,,0000,0000,0000,,And we already worked out B as\Nbeing equal to. Dialogue: 0,0:18:37.94,0:18:42.80,Default,,0000,0000,0000,,For over 3, this means that we\Ncan write out the solution to Dialogue: 0,0:18:42.80,0:18:43.92,Default,,0000,0000,0000,,the whole problem. Dialogue: 0,0:18:44.48,0:18:51.70,Default,,0000,0000,0000,,3X plus one over\NX minus one squared Dialogue: 0,0:18:51.70,0:18:54.41,Default,,0000,0000,0000,,times X +2. Dialogue: 0,0:18:55.00,0:19:02.35,Default,,0000,0000,0000,,Equals. A5 over 9X\Nminus one plus B is Dialogue: 0,0:19:02.35,0:19:09.02,Default,,0000,0000,0000,,4 over 3 four over\N3X minus 1 squared. Dialogue: 0,0:19:09.76,0:19:16.77,Default,,0000,0000,0000,,See is minus 5 over 9, so\Nwe have minus five over 9 X Dialogue: 0,0:19:16.77,0:19:23.00,Default,,0000,0000,0000,,+2. Another case\Nwe must consider. Dialogue: 0,0:19:23.54,0:19:27.99,Default,,0000,0000,0000,,Is where the denominator\Ncontains a quadratic that can't Dialogue: 0,0:19:27.99,0:19:30.95,Default,,0000,0000,0000,,be factorized as in 5X over. Dialogue: 0,0:19:31.46,0:19:37.80,Default,,0000,0000,0000,,X squared plus X Plus One\NTimes X minus 2. Dialogue: 0,0:19:38.50,0:19:42.03,Default,,0000,0000,0000,,If we to express this in partial\Nfractions, the two denominators Dialogue: 0,0:19:42.03,0:19:46.52,Default,,0000,0000,0000,,are going to be X squared plus X\NPlus One and X minus 2. Dialogue: 0,0:19:47.12,0:19:52.08,Default,,0000,0000,0000,,When the denominator is X\Nsquared plus 6 plus one, we have Dialogue: 0,0:19:52.08,0:19:56.21,Default,,0000,0000,0000,,to consider the possibility that\Nthe numerator can contain a Dialogue: 0,0:19:56.21,0:19:59.92,Default,,0000,0000,0000,,termine ex, because the\Nnumerator would still be of Dialogue: 0,0:19:59.92,0:20:03.23,Default,,0000,0000,0000,,lower degree than the\Ndenominator, and this would Dialogue: 0,0:20:03.23,0:20:08.60,Default,,0000,0000,0000,,still therefore be a proper\Nfraction. So we write a X plus B Dialogue: 0,0:20:08.60,0:20:11.49,Default,,0000,0000,0000,,over X squared plus X plus one. Dialogue: 0,0:20:12.09,0:20:19.16,Default,,0000,0000,0000,,Plus C over X minus two\Nas before. We multiply this out Dialogue: 0,0:20:19.16,0:20:26.23,Default,,0000,0000,0000,,so we get that five X\Nequals X plus B Times X Dialogue: 0,0:20:26.23,0:20:33.45,Default,,0000,0000,0000,,minus 2. Plus\NC Times Dialogue: 0,0:20:33.45,0:20:40.01,Default,,0000,0000,0000,,X squared.\NPlus 6 + 1. Dialogue: 0,0:20:41.63,0:20:43.92,Default,,0000,0000,0000,,One special value we can use is Dialogue: 0,0:20:43.92,0:20:46.51,Default,,0000,0000,0000,,X equals 2. And if. Dialogue: 0,0:20:47.02,0:20:50.74,Default,,0000,0000,0000,,X equals 2, we Dialogue: 0,0:20:50.74,0:20:57.00,Default,,0000,0000,0000,,get 5X5210. This is 0,\Nso this all disappears and we Dialogue: 0,0:20:57.00,0:21:04.12,Default,,0000,0000,0000,,get 2 twos of 4 + 2 is 6 plus\None is 7, so 10 equals 7 C. Dialogue: 0,0:21:04.94,0:21:08.52,Default,,0000,0000,0000,,Giving C equals 10 over 7. Dialogue: 0,0:21:09.52,0:21:14.20,Default,,0000,0000,0000,,Unfortunately, there's no value\Nfor X would enable us to get rid Dialogue: 0,0:21:14.20,0:21:19.27,Default,,0000,0000,0000,,of C, so we're going to have to\Nuse the technique of equating Dialogue: 0,0:21:19.27,0:21:25.77,Default,,0000,0000,0000,,coefficients. I'll write\Nthis out again. Dialogue: 0,0:21:26.45,0:21:32.43,Default,,0000,0000,0000,,In order\Nto equate Dialogue: 0,0:21:32.43,0:21:38.41,Default,,0000,0000,0000,,coefficients, I'm\Ngoing to Dialogue: 0,0:21:38.41,0:21:44.39,Default,,0000,0000,0000,,have to\Nmultiply this Dialogue: 0,0:21:44.39,0:21:45.88,Default,,0000,0000,0000,,out. Dialogue: 0,0:21:46.90,0:21:50.03,Default,,0000,0000,0000,,X times X is X squared. Dialogue: 0,0:21:51.14,0:21:54.79,Default,,0000,0000,0000,,X times minus two is minus two Dialogue: 0,0:21:54.79,0:22:01.64,Default,,0000,0000,0000,,AX. B times X\Nis BXB times minus two Dialogue: 0,0:22:01.64,0:22:08.74,Default,,0000,0000,0000,,gives us minus 2B Plus\NCX squared Plus CX Plus Dialogue: 0,0:22:08.74,0:22:13.81,Default,,0000,0000,0000,,C. Again, I'm going to collect\Nlike terms. So for instance for Dialogue: 0,0:22:13.81,0:22:15.19,Default,,0000,0000,0000,,X squared we have. Dialogue: 0,0:22:16.38,0:22:23.09,Default,,0000,0000,0000,,AX squared and CX squared.\NSo we have a plus Dialogue: 0,0:22:23.09,0:22:29.80,Default,,0000,0000,0000,,CX squared for X. We\Nhave a minus two AAB&C. Dialogue: 0,0:22:30.31,0:22:38.20,Default,,0000,0000,0000,,So minus two A+B Plus\NCX and for a constant Dialogue: 0,0:22:38.20,0:22:42.14,Default,,0000,0000,0000,,we have minus 2B Plus Dialogue: 0,0:22:42.14,0:22:48.48,Default,,0000,0000,0000,,C. We still\Nneed to find Dialogue: 0,0:22:48.48,0:22:50.61,Default,,0000,0000,0000,,both A&B. Dialogue: 0,0:22:51.87,0:22:55.73,Default,,0000,0000,0000,,For two unknowns we need 2\Nequations, so we are going to Dialogue: 0,0:22:55.73,0:22:57.67,Default,,0000,0000,0000,,have to solve for two different Dialogue: 0,0:22:57.67,0:23:02.44,Default,,0000,0000,0000,,coefficients. Now the left hand\Nside is just 5X, so there is no Dialogue: 0,0:23:02.44,0:23:03.67,Default,,0000,0000,0000,,coefficient in X squared. Dialogue: 0,0:23:04.28,0:23:09.94,Default,,0000,0000,0000,,In order to eliminate X squared,\Nwe can say that a plus C equals Dialogue: 0,0:23:09.94,0:23:17.46,Default,,0000,0000,0000,,0. We already know what see is\N10 over 7. In order for a plus C Dialogue: 0,0:23:17.46,0:23:22.01,Default,,0000,0000,0000,,to be 0, this will make a minus\N10 over 7. Dialogue: 0,0:23:24.28,0:23:28.70,Default,,0000,0000,0000,,The left hand side also has\Nno constant coefficient, so Dialogue: 0,0:23:28.70,0:23:33.12,Default,,0000,0000,0000,,that means that this\Nexpression must be 0. So we Dialogue: 0,0:23:33.12,0:23:36.21,Default,,0000,0000,0000,,say minus 2B Plus C equals 0. Dialogue: 0,0:23:37.62,0:23:40.88,Default,,0000,0000,0000,,Giving us. C equals Dialogue: 0,0:23:40.88,0:23:48.10,Default,,0000,0000,0000,,2B. Or B equals C over\Ntwo, which gives us B as being. Dialogue: 0,0:23:48.61,0:23:51.08,Default,,0000,0000,0000,,5 over 7. Dialogue: 0,0:23:52.96,0:24:00.35,Default,,0000,0000,0000,,So we have a equal to\Nminus 10 over 7B equal to Dialogue: 0,0:24:00.35,0:24:06.51,Default,,0000,0000,0000,,five over 7 and C equal\Nto 10 over 7. Dialogue: 0,0:24:08.00,0:24:11.20,Default,,0000,0000,0000,,This means that 5X over. Dialogue: 0,0:24:11.70,0:24:15.13,Default,,0000,0000,0000,,X squared plus X plus one. Dialogue: 0,0:24:15.67,0:24:23.09,Default,,0000,0000,0000,,Times X minus two is equal to\Na X which is minus 10 over Dialogue: 0,0:24:23.09,0:24:30.93,Default,,0000,0000,0000,,7X. Plus B, which is 5\Nover 7 all over X squared plus Dialogue: 0,0:24:30.93,0:24:32.58,Default,,0000,0000,0000,,X plus one. Dialogue: 0,0:24:33.78,0:24:40.89,Default,,0000,0000,0000,,Plus C, which is 10 over 7\Nover X minus two and are now Dialogue: 0,0:24:40.89,0:24:47.50,Default,,0000,0000,0000,,tidy. This up the Seven comes\Ndown to be multiplied by the X Dialogue: 0,0:24:47.50,0:24:54.61,Default,,0000,0000,0000,,squared plus X plus one. So we\Nget minus 10X plus five over 7 Dialogue: 0,0:24:54.61,0:25:01.21,Default,,0000,0000,0000,,X squared plus X Plus One plus\Nand again the Seven comes down Dialogue: 0,0:25:01.21,0:25:03.75,Default,,0000,0000,0000,,10 over 7X minus 2. Dialogue: 0,0:25:03.86,0:25:10.20,Default,,0000,0000,0000,,Equals and to finish it off we\Nneed to take five out of this Dialogue: 0,0:25:10.20,0:25:15.18,Default,,0000,0000,0000,,expression as a factor, which\Ngives us five times minus 2X Dialogue: 0,0:25:15.18,0:25:22.43,Default,,0000,0000,0000,,plus one over 7 X squared plus X\Nplus 1 + 10 over 7X minus 2. Dialogue: 0,0:25:23.30,0:25:29.16,Default,,0000,0000,0000,,So far I've only dealt with\Nproper fractions where the Dialogue: 0,0:25:29.16,0:25:35.02,Default,,0000,0000,0000,,numerator is of lower degree\Nthan the denominator. Now, like Dialogue: 0,0:25:35.02,0:25:38.54,Default,,0000,0000,0000,,to look at an improper fraction. Dialogue: 0,0:25:39.22,0:25:40.95,Default,,0000,0000,0000,,Let's Express. Dialogue: 0,0:25:42.17,0:25:49.20,Default,,0000,0000,0000,,4X cubed plus 10X\Nplus four over X Dialogue: 0,0:25:49.20,0:25:52.72,Default,,0000,0000,0000,,into 2X plus one. Dialogue: 0,0:25:53.39,0:25:55.36,Default,,0000,0000,0000,,In partial fractions. Dialogue: 0,0:25:57.45,0:25:59.46,Default,,0000,0000,0000,,The numerator is of degree 3. Dialogue: 0,0:26:00.59,0:26:05.10,Default,,0000,0000,0000,,The denominator, if you multiply\Nthe X by the two X, you get 2 X Dialogue: 0,0:26:05.10,0:26:06.61,Default,,0000,0000,0000,,squared, so the denominator is Dialogue: 0,0:26:06.61,0:26:10.58,Default,,0000,0000,0000,,of degree 2. This means that\Nthis is an improper fraction. Dialogue: 0,0:26:11.75,0:26:15.100,Default,,0000,0000,0000,,What this means is that if you\Ndivide the numerator by the Dialogue: 0,0:26:15.100,0:26:19.89,Default,,0000,0000,0000,,denominator, you're going to be\Ndividing otermin X cubed by a Dialogue: 0,0:26:19.89,0:26:21.31,Default,,0000,0000,0000,,term in X squared. Dialogue: 0,0:26:22.18,0:26:24.64,Default,,0000,0000,0000,,So you could get a Terminix. Dialogue: 0,0:26:25.16,0:26:29.79,Default,,0000,0000,0000,,Which means that we have to\Nwrite down acts. We may also get Dialogue: 0,0:26:29.79,0:26:33.35,Default,,0000,0000,0000,,a constant term, so we have to\Nwrite down B. Dialogue: 0,0:26:33.91,0:26:36.28,Default,,0000,0000,0000,,Then we can do our fractions. Dialogue: 0,0:26:37.13,0:26:44.51,Default,,0000,0000,0000,,If I now multiply but\Nthrough I get a X Dialogue: 0,0:26:44.51,0:26:51.89,Default,,0000,0000,0000,,Times X Times 2X plus\None, so we get 4X Dialogue: 0,0:26:51.89,0:26:55.58,Default,,0000,0000,0000,,cubed plus 10X plus four Dialogue: 0,0:26:55.58,0:26:58.85,Default,,0000,0000,0000,,equals a. X squared Dialogue: 0,0:26:59.41,0:27:02.56,Default,,0000,0000,0000,,Times 2X plus one. Dialogue: 0,0:27:02.56,0:27:06.25,Default,,0000,0000,0000,,Plus BX times 2X Dialogue: 0,0:27:06.25,0:27:13.21,Default,,0000,0000,0000,,plus one. Plus C\NTimes 2X plus one. Dialogue: 0,0:27:14.50,0:27:17.67,Default,,0000,0000,0000,,Plus DX Dialogue: 0,0:27:21.53,0:27:23.04,Default,,0000,0000,0000,,Using special values. Dialogue: 0,0:27:23.73,0:27:29.46,Default,,0000,0000,0000,,If I use X equals 0, then\Nthe term the D, the B, and Dialogue: 0,0:27:29.46,0:27:33.96,Default,,0000,0000,0000,,the A are all going to\Ndisappear and I'm just left Dialogue: 0,0:27:33.96,0:27:36.82,Default,,0000,0000,0000,,with see. So if X equals 0. Dialogue: 0,0:27:37.93,0:27:42.68,Default,,0000,0000,0000,,X cubed is zero, X is zero. I\Njust get 4 equal to. Dialogue: 0,0:27:44.37,0:27:50.71,Default,,0000,0000,0000,,2X is 0, so it's just C, so we\Nhave C equal to four. The other Dialogue: 0,0:27:50.71,0:27:53.08,Default,,0000,0000,0000,,special value is X equal to Dialogue: 0,0:27:53.08,0:27:59.39,Default,,0000,0000,0000,,minus 1/2. If X equals minus\NAlpha, this is 0, so this will Dialogue: 0,0:27:59.39,0:28:02.40,Default,,0000,0000,0000,,disappear. This is 0, so this Dialogue: 0,0:28:02.40,0:28:06.70,Default,,0000,0000,0000,,will disappear. And this will\Ndisappear, just leaving me with Dialogue: 0,0:28:06.70,0:28:09.32,Default,,0000,0000,0000,,D. So I get. Dialogue: 0,0:28:10.33,0:28:15.26,Default,,0000,0000,0000,,Minus 1/2. Cubed is\Nminus an eighth, so we Dialogue: 0,0:28:15.26,0:28:17.51,Default,,0000,0000,0000,,get minus four over 8. Dialogue: 0,0:28:18.65,0:28:25.92,Default,,0000,0000,0000,,Plus 10 times minus 1/2 inches\Nminus 10 over 2 + 4 Dialogue: 0,0:28:25.92,0:28:29.38,Default,,0000,0000,0000,,equals. D times minus Dialogue: 0,0:28:29.38,0:28:36.48,Default,,0000,0000,0000,,1/2. I'll just\Nwrite that down again, Dialogue: 0,0:28:36.48,0:28:44.18,Default,,0000,0000,0000,,minus four over 8\N- 10 over 2. Dialogue: 0,0:28:45.48,0:28:49.34,Default,,0000,0000,0000,,+4. Equals minus Dialogue: 0,0:28:49.34,0:28:54.70,Default,,0000,0000,0000,,1/2 D. Minus 4 over 8\Nis just minus 1/2. Dialogue: 0,0:28:55.29,0:29:02.44,Default,,0000,0000,0000,,Minus 10 over 2 is minus 5\N+ 4 equals minus half D. Dialogue: 0,0:29:04.29,0:29:11.58,Default,,0000,0000,0000,,Minus 5 + 4 is minus one,\Nso I've got minus 1 1/2 equals Dialogue: 0,0:29:11.58,0:29:13.15,Default,,0000,0000,0000,,minus 1/2 D. Dialogue: 0,0:29:16.23,0:29:21.43,Default,,0000,0000,0000,,Minus 1 1/2 is just three\Ntimes minus 1/2, so this Dialogue: 0,0:29:21.43,0:29:23.80,Default,,0000,0000,0000,,gives us D equal 3. Dialogue: 0,0:29:25.09,0:29:31.05,Default,,0000,0000,0000,,Special values won't give me a\Nor be, so I'm going to have to Dialogue: 0,0:29:31.05,0:29:35.31,Default,,0000,0000,0000,,equate coefficients. This means\NI have to write this expression Dialogue: 0,0:29:35.31,0:29:40.88,Default,,0000,0000,0000,,out again. 4X\Ncubed plus Dialogue: 0,0:29:40.88,0:29:48.20,Default,,0000,0000,0000,,10X. +4 equals\Na X squared times. Dialogue: 0,0:29:48.77,0:29:55.22,Default,,0000,0000,0000,,2X plus one plus\NBX times 2X plus Dialogue: 0,0:29:55.22,0:29:57.23,Default,,0000,0000,0000,,one. Plus Dialogue: 0,0:29:58.26,0:30:03.86,Default,,0000,0000,0000,,C times 2X plus one\Nplus DX. Dialogue: 0,0:30:05.55,0:30:07.50,Default,,0000,0000,0000,,I'm now going to multiply\Nthis out. Dialogue: 0,0:30:08.94,0:30:13.86,Default,,0000,0000,0000,,X squared times\N2X is 2A X cubed. Dialogue: 0,0:30:14.92,0:30:18.86,Default,,0000,0000,0000,,X squared times one\Nis just X squared. Dialogue: 0,0:30:19.96,0:30:22.44,Default,,0000,0000,0000,,This gives me 2B X squared. Dialogue: 0,0:30:24.85,0:30:27.96,Default,,0000,0000,0000,,This gives me Dialogue: 0,0:30:27.96,0:30:31.58,Default,,0000,0000,0000,,BX. This gives Me 2 Dialogue: 0,0:30:31.58,0:30:36.82,Default,,0000,0000,0000,,CX. This gives\Nme C. Dialogue: 0,0:30:37.99,0:30:38.66,Default,,0000,0000,0000,,And then. Dialogue: 0,0:30:39.84,0:30:47.33,Default,,0000,0000,0000,,Plus DX And collecting terms, we\Nonly have one Turman X cubed, so Dialogue: 0,0:30:47.33,0:30:50.33,Default,,0000,0000,0000,,that is just 2A X cubed. Dialogue: 0,0:30:51.01,0:30:55.99,Default,,0000,0000,0000,,Plus we have two terms in X\Nsquared, A and 2B. Dialogue: 0,0:30:56.83,0:31:04.55,Default,,0000,0000,0000,,We have three terms\Nin XB2C and D. Dialogue: 0,0:31:05.12,0:31:10.63,Default,,0000,0000,0000,,And finally,\Nthe constant Dialogue: 0,0:31:10.63,0:31:13.39,Default,,0000,0000,0000,,term see. Dialogue: 0,0:31:15.05,0:31:19.57,Default,,0000,0000,0000,,Now look at the Turman X cubed.\NWe have 4X cubed on the left. Dialogue: 0,0:31:20.33,0:31:25.86,Default,,0000,0000,0000,,And two AX cubed on the right.\NThis means that 2A must be equal Dialogue: 0,0:31:25.86,0:31:32.28,Default,,0000,0000,0000,,to 4. Giving us a equal to two\Nnow look at the Turman X Dialogue: 0,0:31:32.28,0:31:35.85,Default,,0000,0000,0000,,squared. There is no Turman X\Nsquared on the left. Dialogue: 0,0:31:36.60,0:31:39.23,Default,,0000,0000,0000,,And on the right\Nwe have a plus 2B. Dialogue: 0,0:31:40.31,0:31:43.73,Default,,0000,0000,0000,,This means that as there isn't\NTurman X squared on the left, a Dialogue: 0,0:31:43.73,0:31:45.57,Default,,0000,0000,0000,,plus 2B must be equal to 0. Dialogue: 0,0:31:46.12,0:31:49.65,Default,,0000,0000,0000,,So we have a plus 2B Dialogue: 0,0:31:49.65,0:31:52.82,Default,,0000,0000,0000,,equals 0. Which means that. Dialogue: 0,0:31:53.45,0:31:56.95,Default,,0000,0000,0000,,A equals minus Dialogue: 0,0:31:56.95,0:32:01.56,Default,,0000,0000,0000,,2B. Which means\Nthat B equals. Dialogue: 0,0:32:03.04,0:32:07.78,Default,,0000,0000,0000,,Minus two over 2\Nequals minus one. Dialogue: 0,0:32:09.11,0:32:15.09,Default,,0000,0000,0000,,I'll just write those\Nvalues in again. Dialogue: 0,0:32:15.80,0:32:17.77,Default,,0000,0000,0000,,A equals 2. Dialogue: 0,0:32:18.51,0:32:24.94,Default,,0000,0000,0000,,B equals minus one C\Nequals 4D equals 3. Dialogue: 0,0:32:27.70,0:32:34.12,Default,,0000,0000,0000,,So if we take our original\Nexpression 4X cubed plus 10X Dialogue: 0,0:32:34.12,0:32:37.04,Default,,0000,0000,0000,,plus four over X times. Dialogue: 0,0:32:38.89,0:32:44.81,Default,,0000,0000,0000,,2X plus one. This is equal\Nto axe, so 2X. Dialogue: 0,0:32:45.39,0:32:52.43,Default,,0000,0000,0000,,Minus B. Plus see over X,\Nso that's four over X Plus D Dialogue: 0,0:32:52.43,0:32:57.28,Default,,0000,0000,0000,,over 2X Plus One which is 3 over\N2X plus one.