[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.29,0:00:05.66,Default,,0000,0000,0000,,The general strategy for solving\Na cubic equation is to reduce it
Dialogue: 0,0:00:05.66,0:00:10.03,Default,,0000,0000,0000,,to a quadratic and then solve\Nthe quadratic by the usual means
Dialogue: 0,0:00:10.03,0:00:11.85,Default,,0000,0000,0000,,either by Factorizing or using
Dialogue: 0,0:00:11.85,0:00:18.100,Default,,0000,0000,0000,,the formula. A cubic equation\Nhas the form a X cubed
Dialogue: 0,0:00:18.100,0:00:24.71,Default,,0000,0000,0000,,plus BX squared plus CX plus\ND equals not.
Dialogue: 0,0:00:25.28,0:00:28.67,Default,,0000,0000,0000,,It must have a Turman X cubed,\Nor it wouldn't be a cubic.
Dialogue: 0,0:00:29.35,0:00:33.29,Default,,0000,0000,0000,,But any or all of BC&D can be 0.
Dialogue: 0,0:00:34.89,0:00:37.63,Default,,0000,0000,0000,,So for instance X cubed.
Dialogue: 0,0:00:39.01,0:00:45.36,Default,,0000,0000,0000,,Minus six X squared plus\N11X minus six equals note
Dialogue: 0,0:00:45.36,0:00:47.26,Default,,0000,0000,0000,,is a cubic.
Dialogue: 0,0:00:48.53,0:00:53.58,Default,,0000,0000,0000,,So is 4X cubed plus 57\Nequals not.
Dialogue: 0,0:00:55.62,0:00:58.97,Default,,0000,0000,0000,,So is. X cubed
Dialogue: 0,0:00:59.56,0:01:03.04,Default,,0000,0000,0000,,+9 X equals not.
Dialogue: 0,0:01:04.95,0:01:08.77,Default,,0000,0000,0000,,Just as a quadratic equation may\Nhave two real roots.
Dialogue: 0,0:01:09.58,0:01:12.28,Default,,0000,0000,0000,,So a cubic equation\Npossibly has three.
Dialogue: 0,0:01:13.29,0:01:17.51,Default,,0000,0000,0000,,But unlike a quadratic equation\Nwhich main have no real
Dialogue: 0,0:01:17.51,0:01:21.73,Default,,0000,0000,0000,,solution, a cubic equation\Nalways has at least one real
Dialogue: 0,0:01:21.73,0:01:23.84,Default,,0000,0000,0000,,root. I'll explain why later.
Dialogue: 0,0:01:24.74,0:01:28.59,Default,,0000,0000,0000,,If a cubic does have 3\NRoutes, 2 or even all three
Dialogue: 0,0:01:28.59,0:01:30.20,Default,,0000,0000,0000,,of them may be repeated.
Dialogue: 0,0:01:31.43,0:01:37.79,Default,,0000,0000,0000,,This gives\Nus four
Dialogue: 0,0:01:37.79,0:01:39.38,Default,,0000,0000,0000,,possibilities.
Dialogue: 0,0:01:41.41,0:01:48.96,Default,,0000,0000,0000,,The equation X cubed minus\Nsix X squared plus 11X.
Dialogue: 0,0:01:49.85,0:01:56.70,Default,,0000,0000,0000,,Minus 6 equals 0 Factorizes\N2X minus one times X
Dialogue: 0,0:01:56.70,0:02:02.18,Default,,0000,0000,0000,,minus two times X minus\Nthree equals 0.
Dialogue: 0,0:02:02.99,0:02:06.11,Default,,0000,0000,0000,,This equation has three real\Nroots, all different.
Dialogue: 0,0:02:06.84,0:02:11.35,Default,,0000,0000,0000,,Solutions X equals 1 X equals 2\Nor X equals 3.
Dialogue: 0,0:02:11.89,0:02:16.83,Default,,0000,0000,0000,,I'd like to show you the graph\Nof the curve Y equals X cubed
Dialogue: 0,0:02:16.83,0:02:21.42,Default,,0000,0000,0000,,minus six X squared plus 11X\Nminus six. I'm not very good at
Dialogue: 0,0:02:21.42,0:02:22.48,Default,,0000,0000,0000,,drawing freehand graphs.
Dialogue: 0,0:02:23.01,0:02:25.14,Default,,0000,0000,0000,,So here's one I prepared\Nearlier.
Dialogue: 0,0:02:27.98,0:02:33.15,Default,,0000,0000,0000,,Notice that it starts slow down\Nto the left.
Dialogue: 0,0:02:33.86,0:02:37.98,Default,,0000,0000,0000,,Because as X gets larger,\Nnegative, so does X cubed and it
Dialogue: 0,0:02:37.98,0:02:41.75,Default,,0000,0000,0000,,finishes high to the right\Nbecause there's X gets large and
Dialogue: 0,0:02:41.75,0:02:43.46,Default,,0000,0000,0000,,positive. So does X cubed.
Dialogue: 0,0:02:44.42,0:02:48.94,Default,,0000,0000,0000,,And the curve crosses the X axis\Nthree times, once where X equals
Dialogue: 0,0:02:48.94,0:02:53.47,Default,,0000,0000,0000,,1 once, where X equals 2 and\Nonce where X equals 3. This
Dialogue: 0,0:02:53.47,0:02:55.21,Default,,0000,0000,0000,,gives us our three separate
Dialogue: 0,0:02:55.21,0:03:01.86,Default,,0000,0000,0000,,solutions. The\Nequation X
Dialogue: 0,0:03:01.86,0:03:05.79,Default,,0000,0000,0000,,cubed. Minus five
Dialogue: 0,0:03:05.79,0:03:11.40,Default,,0000,0000,0000,,X squared. Plus 8X\Nminus 4 equals 0
Dialogue: 0,0:03:11.40,0:03:16.80,Default,,0000,0000,0000,,Factorizes 2X minus one\Ntimes X minus two all
Dialogue: 0,0:03:16.80,0:03:21.01,Default,,0000,0000,0000,,squared, and that is\Nequal to 0.
Dialogue: 0,0:03:22.63,0:03:26.77,Default,,0000,0000,0000,,In this case we have. We do have\N3 routes, but two of them are
Dialogue: 0,0:03:26.77,0:03:30.08,Default,,0000,0000,0000,,the same. We have X minus 2\Nsquared, so we only actually
Dialogue: 0,0:03:30.08,0:03:35.55,Default,,0000,0000,0000,,have two solutions. Again, I'll\Nshow you the graph of Y equals X
Dialogue: 0,0:03:35.55,0:03:39.82,Default,,0000,0000,0000,,cubed minus five X squared plus\N8X minus 40 equals 0.
Dialogue: 0,0:03:45.47,0:03:49.50,Default,,0000,0000,0000,,Again, the curve starts load\Nto the left and goes high to
Dialogue: 0,0:03:49.50,0:03:53.20,Default,,0000,0000,0000,,the right. It crosses the X\Naxis once and then just
Dialogue: 0,0:03:53.20,0:03:57.90,Default,,0000,0000,0000,,touches it. So we have our two\Nroots, X equals 1 and X equals
Dialogue: 0,0:03:57.90,0:04:01.60,Default,,0000,0000,0000,,2 and it touches at the\Nrepeated root X equals 2.
Dialogue: 0,0:04:03.22,0:04:10.69,Default,,0000,0000,0000,,The equation X cubed\Nminus three X squared
Dialogue: 0,0:04:10.69,0:04:14.43,Default,,0000,0000,0000,,plus 3X minus one
Dialogue: 0,0:04:14.43,0:04:20.90,Default,,0000,0000,0000,,equals note. The left\Nhand side factorizes to X minus
Dialogue: 0,0:04:20.90,0:04:23.30,Default,,0000,0000,0000,,one or cubed equals not.
Dialogue: 0,0:04:23.83,0:04:27.22,Default,,0000,0000,0000,,So there there are three\Nfactors. They're all the same
Dialogue: 0,0:04:27.22,0:04:30.61,Default,,0000,0000,0000,,and we only have a single\Nsolution. X equals 1.
Dialogue: 0,0:04:31.84,0:04:36.06,Default,,0000,0000,0000,,The corresponding curve is Y\Nequals X cubed, minus three X
Dialogue: 0,0:04:36.06,0:04:37.98,Default,,0000,0000,0000,,squared plus 3X minus one.
Dialogue: 0,0:04:38.70,0:04:41.35,Default,,0000,0000,0000,,And it looks like this.
Dialogue: 0,0:04:44.87,0:04:47.74,Default,,0000,0000,0000,,As with all the Cubix I've\Nshown you so far, it starts
Dialogue: 0,0:04:47.74,0:04:50.61,Default,,0000,0000,0000,,slow down on the left and\Ngoes high up to the right.
Dialogue: 0,0:04:51.96,0:04:55.96,Default,,0000,0000,0000,,Notice that the curve does\Ncross the X axis at the point X
Dialogue: 0,0:04:55.96,0:05:00.28,Default,,0000,0000,0000,,equals 1, but it is also a\Ntangent. X axis is a tangent to
Dialogue: 0,0:05:00.28,0:05:03.05,Default,,0000,0000,0000,,the curve at this point,\Nindicating the three repeated
Dialogue: 0,0:05:03.05,0:05:03.36,Default,,0000,0000,0000,,roots.
Dialogue: 0,0:05:06.80,0:05:14.78,Default,,0000,0000,0000,,Now look at the equation\NX cubed plus X squared
Dialogue: 0,0:05:14.78,0:05:19.95,Default,,0000,0000,0000,,plus X. Minus\N3 equals 0.
Dialogue: 0,0:05:21.00,0:05:23.79,Default,,0000,0000,0000,,This expression Factorizes 2X
Dialogue: 0,0:05:23.79,0:05:31.42,Default,,0000,0000,0000,,minus one. X squared plus 2X\Nplus three, so we can put this
Dialogue: 0,0:05:31.42,0:05:32.97,Default,,0000,0000,0000,,equal to 0.
Dialogue: 0,0:05:35.08,0:05:40.79,Default,,0000,0000,0000,,The quadratic X squared +2 X +3\Nequals not has no real
Dialogue: 0,0:05:40.79,0:05:44.43,Default,,0000,0000,0000,,solutions. So the only\Nsolution to the cubic
Dialogue: 0,0:05:44.43,0:05:48.96,Default,,0000,0000,0000,,equation is to put X minus\None equal to 0, giving this
Dialogue: 0,0:05:48.96,0:05:51.23,Default,,0000,0000,0000,,single real solution X equals\N1.
Dialogue: 0,0:05:53.40,0:05:59.72,Default,,0000,0000,0000,,The graph Y equals X cubed plus\NX squared plus 6 - 3 looks like
Dialogue: 0,0:05:59.72,0:06:07.29,Default,,0000,0000,0000,,this. And you can\Nsee that it only
Dialogue: 0,0:06:07.29,0:06:13.77,Default,,0000,0000,0000,,crosses the X axis\Nin one place.
Dialogue: 0,0:06:14.38,0:06:16.01,Default,,0000,0000,0000,,From the graphs that I've shown
Dialogue: 0,0:06:16.01,0:06:19.94,Default,,0000,0000,0000,,you. You can see why a cubic\Nequation always has at least
Dialogue: 0,0:06:19.94,0:06:20.79,Default,,0000,0000,0000,,one real root.
Dialogue: 0,0:06:22.11,0:06:25.66,Default,,0000,0000,0000,,The graph either starts large,\Nnegative, and finishes large
Dialogue: 0,0:06:25.66,0:06:30.38,Default,,0000,0000,0000,,positive. If the coefficient of\NX cubed is positive or it will
Dialogue: 0,0:06:30.38,0:06:33.54,Default,,0000,0000,0000,,start large positive and\Nfinished down here. Large
Dialogue: 0,0:06:33.54,0:06:38.26,Default,,0000,0000,0000,,negative if the coefficient of X\Ncubed is negative, the graph of
Dialogue: 0,0:06:38.26,0:06:43.78,Default,,0000,0000,0000,,a cubic must cross the X axis,\Ngiving you one real root. So any
Dialogue: 0,0:06:43.78,0:06:47.72,Default,,0000,0000,0000,,problem you get that involves\Nsolving a cubic equation will
Dialogue: 0,0:06:47.72,0:06:49.30,Default,,0000,0000,0000,,have a real solution.
Dialogue: 0,0:06:52.72,0:06:59.06,Default,,0000,0000,0000,,Now let's move on to\Nhow we solve cubics.
Dialogue: 0,0:06:59.09,0:07:05.31,Default,,0000,0000,0000,,Like a quadratic, cubic should\Nalways be rearranged into the
Dialogue: 0,0:07:05.31,0:07:12.77,Default,,0000,0000,0000,,form X cubed plus BX squared\Nplus CX plus D equals 0.
Dialogue: 0,0:07:13.75,0:07:20.44,Default,,0000,0000,0000,,The equation X squared plus 4X\Nminus one equals 6 over X is a
Dialogue: 0,0:07:20.44,0:07:26.18,Default,,0000,0000,0000,,cubic, but I wouldn't like to\Ntry and solve it in this
Dialogue: 0,0:07:26.18,0:07:31.44,Default,,0000,0000,0000,,particular form. We need to\Nmultiply through by X, giving us
Dialogue: 0,0:07:31.44,0:07:36.22,Default,,0000,0000,0000,,X cubed plus four X squared\Nminus X equals 6.
Dialogue: 0,0:07:36.74,0:07:40.94,Default,,0000,0000,0000,,And then we subtract 6 from both\Nsides, giving us X cubed.
Dialogue: 0,0:07:41.54,0:07:48.25,Default,,0000,0000,0000,,Plus four X squared minus\NX minus 6 equals 0.
Dialogue: 0,0:07:49.88,0:07:54.74,Default,,0000,0000,0000,,When solving cubics, it\Nhelps if you know or
Dialogue: 0,0:07:54.74,0:07:59.60,Default,,0000,0000,0000,,think you know one route\Nto start with. For
Dialogue: 0,0:07:59.60,0:08:02.84,Default,,0000,0000,0000,,instance, take the\Nequation X cubed.
Dialogue: 0,0:08:04.76,0:08:11.30,Default,,0000,0000,0000,,Minus five X squared\Nminus 2X plus 24
Dialogue: 0,0:08:11.30,0:08:18.19,Default,,0000,0000,0000,,equals not. Given that\NX equals minus two is a
Dialogue: 0,0:08:18.19,0:08:22.38,Default,,0000,0000,0000,,solution. There is a theorem\Ncalled the factor theorem which
Dialogue: 0,0:08:22.38,0:08:26.33,Default,,0000,0000,0000,,I'm not going to attempt to\Nprove here that says that if X
Dialogue: 0,0:08:26.33,0:08:28.15,Default,,0000,0000,0000,,equals minus two is a solution
Dialogue: 0,0:08:28.15,0:08:35.41,Default,,0000,0000,0000,,of this equation. Then X\N+2 is a
Dialogue: 0,0:08:35.41,0:08:39.35,Default,,0000,0000,0000,,factor. Of this whole\Nexpression.
Dialogue: 0,0:08:40.59,0:08:47.08,Default,,0000,0000,0000,,This means that X cubed minus\Nfive X squared minus 2X plus 24
Dialogue: 0,0:08:47.08,0:08:49.57,Default,,0000,0000,0000,,is equal to X +2.
Dialogue: 0,0:08:50.24,0:08:56.14,Default,,0000,0000,0000,,Times some quadratic which\Nwill call X squared plus
Dialogue: 0,0:08:56.14,0:08:58.10,Default,,0000,0000,0000,,8X Plus B.
Dialogue: 0,0:08:59.22,0:09:01.20,Default,,0000,0000,0000,,And then all that is\Nequal to, not.
Dialogue: 0,0:09:03.06,0:09:08.50,Default,,0000,0000,0000,,So our task now is to find A&B\Nand we do this by a process
Dialogue: 0,0:09:08.50,0:09:09.59,Default,,0000,0000,0000,,called synthetic division.
Dialogue: 0,0:09:12.44,0:09:15.65,Default,,0000,0000,0000,,This involves looking\Nat the coefficients of.
Dialogue: 0,0:09:16.68,0:09:18.17,Default,,0000,0000,0000,,The original expression.
Dialogue: 0,0:09:19.55,0:09:22.61,Default,,0000,0000,0000,,So for instance, the coefficient\Nof X cubed is one.
Dialogue: 0,0:09:23.19,0:09:28.57,Default,,0000,0000,0000,,The coefficient of X squared\Nis minus 5. The coefficient
Dialogue: 0,0:09:28.57,0:09:31.26,Default,,0000,0000,0000,,of X is minus 2.
Dialogue: 0,0:09:32.37,0:09:34.87,Default,,0000,0000,0000,,And the constant is 24.
Dialogue: 0,0:09:35.65,0:09:38.13,Default,,0000,0000,0000,,And we just right in\Nthat we're
Dialogue: 0,0:09:38.13,0:09:39.90,Default,,0000,0000,0000,,synthetically dividing\Nby minus 2.
Dialogue: 0,0:09:41.62,0:09:43.06,Default,,0000,0000,0000,,I leave a line.
Dialogue: 0,0:09:44.74,0:09:51.02,Default,,0000,0000,0000,,And then bring the one down one\Ntimes minus two is minus 2.
Dialogue: 0,0:09:52.21,0:09:55.17,Default,,0000,0000,0000,,Minus 5 plus minus 2.
Dialogue: 0,0:09:55.83,0:09:57.04,Default,,0000,0000,0000,,Is minus 7.
Dialogue: 0,0:09:58.19,0:10:01.18,Default,,0000,0000,0000,,Minus Seven times minus two is
Dialogue: 0,0:10:01.18,0:10:04.83,Default,,0000,0000,0000,,14. 14 plus minus two
Dialogue: 0,0:10:04.83,0:10:08.44,Default,,0000,0000,0000,,is 12. 12 times minus
Dialogue: 0,0:10:08.44,0:10:11.71,Default,,0000,0000,0000,,2. Is minus 24?
Dialogue: 0,0:10:12.22,0:10:15.92,Default,,0000,0000,0000,,And 24 plus minus 24 is 0.
Dialogue: 0,0:10:16.71,0:10:22.42,Default,,0000,0000,0000,,The zero tells us that X +2 is\Nindeed a factor, and the numbers
Dialogue: 0,0:10:22.42,0:10:26.91,Default,,0000,0000,0000,,we have here give us the\Ncoefficients of the quadratic. A
Dialogue: 0,0:10:26.91,0:10:32.62,Default,,0000,0000,0000,,is equal to minus Seven and B is\Nequal to 12. So the quadratic
Dialogue: 0,0:10:32.62,0:10:36.29,Default,,0000,0000,0000,,that we're looking for is X\Nsquared minus 7X.
Dialogue: 0,0:10:37.08,0:10:42.30,Default,,0000,0000,0000,,Plus 12. And synthetic division\Nis explained fully in the
Dialogue: 0,0:10:42.30,0:10:48.41,Default,,0000,0000,0000,,accompanying notes. So we've\Nreduced our cubic 2X plus
Dialogue: 0,0:10:48.41,0:10:55.10,Default,,0000,0000,0000,,two times X squared minus\N7X plus 12 equals zero
Dialogue: 0,0:10:55.10,0:11:01.79,Default,,0000,0000,0000,,X squared minus 7X plus\N12 can be factorized into
Dialogue: 0,0:11:01.79,0:11:08.48,Default,,0000,0000,0000,,X minus three times X\Nminus four. So we have
Dialogue: 0,0:11:08.48,0:11:15.17,Default,,0000,0000,0000,,X +2 times X minus\Nthree times X minus 4.
Dialogue: 0,0:11:15.81,0:11:18.39,Default,,0000,0000,0000,,Equals 0. Giving us.
Dialogue: 0,0:11:19.30,0:11:21.54,Default,,0000,0000,0000,,X equals minus 2.
Dialogue: 0,0:11:22.69,0:11:25.90,Default,,0000,0000,0000,,3 or 4.
Dialogue: 0,0:11:28.33,0:11:35.00,Default,,0000,0000,0000,,If you don't know a\Nroute, it's always worth trying
Dialogue: 0,0:11:35.00,0:11:40.34,Default,,0000,0000,0000,,a few simple values. Let's\Nsolve X cubed.
Dialogue: 0,0:11:40.93,0:11:41.71,Default,,0000,0000,0000,,Minus.
Dialogue: 0,0:11:43.01,0:11:44.22,Default,,0000,0000,0000,,7X.
Dialogue: 0,0:11:45.62,0:11:48.46,Default,,0000,0000,0000,,Minus 6. Equals 0.
Dialogue: 0,0:11:49.04,0:11:51.13,Default,,0000,0000,0000,,The simplest value should try is
Dialogue: 0,0:11:51.13,0:11:53.92,Default,,0000,0000,0000,,one. 1 - 7.
Dialogue: 0,0:11:54.82,0:12:00.46,Default,,0000,0000,0000,,Minus 6 - 12 so that doesn't\Nwork. Let's try minus 1 - 1
Dialogue: 0,0:12:00.46,0:12:06.91,Default,,0000,0000,0000,,cubed is minus 1 + 7 - 6 is\N0, so minus one is a route.
Dialogue: 0,0:12:07.87,0:12:10.44,Default,,0000,0000,0000,,Which means that X plus\None is a factor.
Dialogue: 0,0:12:12.00,0:12:18.10,Default,,0000,0000,0000,,If minus one is a route, we\Ncan synthetically divide through
Dialogue: 0,0:12:18.10,0:12:20.88,Default,,0000,0000,0000,,this expression by minus one.
Dialogue: 0,0:12:21.68,0:12:26.29,Default,,0000,0000,0000,,Coefficient of X squared, sorry\Ncoefficient of X cubed is one.
Dialogue: 0,0:12:27.24,0:12:31.64,Default,,0000,0000,0000,,The coefficient of X squared is\N0, there's no Turman X squared.
Dialogue: 0,0:12:32.60,0:12:38.48,Default,,0000,0000,0000,,The coefficient of X is minus\NSeven and the constant is minus
Dialogue: 0,0:12:38.48,0:12:42.40,Default,,0000,0000,0000,,6 and were synthetically\Ndividing by minus one.
Dialogue: 0,0:12:43.26,0:12:48.13,Default,,0000,0000,0000,,I bring the one down one times\Nminus one is minus one.
Dialogue: 0,0:12:48.65,0:12:50.95,Default,,0000,0000,0000,,Minus one and zero is minus one.
Dialogue: 0,0:12:51.65,0:12:54.03,Default,,0000,0000,0000,,Minus one times minus one is
Dialogue: 0,0:12:54.03,0:12:58.46,Default,,0000,0000,0000,,one. Minus 7 Plus\NOne is minus 6.
Dialogue: 0,0:12:59.51,0:13:06.48,Default,,0000,0000,0000,,Minus six times minus one is 6\N- 6 at 6 is 0.
Dialogue: 0,0:13:10.34,0:13:13.99,Default,,0000,0000,0000,,These numbers give us the\Ncoefficients of the quadratic
Dialogue: 0,0:13:13.99,0:13:16.43,Default,,0000,0000,0000,,equation, and we have X squared.
Dialogue: 0,0:13:17.42,0:13:23.13,Default,,0000,0000,0000,,Minus X minus 6 equals\N0 and we now need to
Dialogue: 0,0:13:23.13,0:13:24.69,Default,,0000,0000,0000,,solve this equation.
Dialogue: 0,0:13:26.40,0:13:29.73,Default,,0000,0000,0000,,So we have X
Dialogue: 0,0:13:29.73,0:13:33.93,Default,,0000,0000,0000,,plus one. Times X\Nsquared.
Dialogue: 0,0:13:35.99,0:13:43.04,Default,,0000,0000,0000,,Minus X minus 6 equal to zero\NX squared minus X minus 6,
Dialogue: 0,0:13:43.04,0:13:50.08,Default,,0000,0000,0000,,factorizes to X minus three X\N+2, so we have X Plus one
Dialogue: 0,0:13:50.08,0:13:56.59,Default,,0000,0000,0000,,times X, minus three times X +2\Nequals 0, and the three
Dialogue: 0,0:13:56.59,0:14:02.55,Default,,0000,0000,0000,,solutions to the cubic equation\NRX equals minus 2 - 1.
Dialogue: 0,0:14:03.09,0:14:04.68,Default,,0000,0000,0000,,Or three.
Dialogue: 0,0:14:06.93,0:14:12.19,Default,,0000,0000,0000,,Sometimes you may be able to\Nspot a factor.
Dialogue: 0,0:14:12.75,0:14:15.48,Default,,0000,0000,0000,,In the equation X cubed.
Dialogue: 0,0:14:16.44,0:14:23.10,Default,,0000,0000,0000,,Minus four X squared\Nminus 9X plus 36
Dialogue: 0,0:14:23.10,0:14:24.76,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,0:14:25.84,0:14:30.26,Default,,0000,0000,0000,,The coefficient of X squared is\Nminus four times the coefficient
Dialogue: 0,0:14:30.26,0:14:31.47,Default,,0000,0000,0000,,of X cubed.
Dialogue: 0,0:14:32.20,0:14:34.79,Default,,0000,0000,0000,,And the constant is minus four
Dialogue: 0,0:14:34.79,0:14:37.48,Default,,0000,0000,0000,,times. The coefficient of X.
Dialogue: 0,0:14:38.85,0:14:43.58,Default,,0000,0000,0000,,This means that we're going to\Nbe able to take out X minus 4 as
Dialogue: 0,0:14:43.58,0:14:47.04,Default,,0000,0000,0000,,a factor from that those two\Nterms and those two terms.
Dialogue: 0,0:14:47.83,0:14:52.86,Default,,0000,0000,0000,,Those two terms are X squared\Ntimes X minus four X squared
Dialogue: 0,0:14:52.86,0:14:57.89,Default,,0000,0000,0000,,times, X is X cubed X squared\Ntimes minus four is minus
Dialogue: 0,0:14:57.89,0:14:59.14,Default,,0000,0000,0000,,four X squared.
Dialogue: 0,0:15:00.36,0:15:03.09,Default,,0000,0000,0000,,Those two terms are minus.
Dialogue: 0,0:15:04.51,0:15:08.40,Default,,0000,0000,0000,,9 times X minus 4.
Dialogue: 0,0:15:10.40,0:15:14.56,Default,,0000,0000,0000,,Minus nine times X is minus\N9X and minus nine times minus
Dialogue: 0,0:15:14.56,0:15:18.38,Default,,0000,0000,0000,,four is plus 36 and all that\Nis equal to 0.
Dialogue: 0,0:15:20.13,0:15:25.37,Default,,0000,0000,0000,,And we can now factorize again\Nbecause we have a common factor
Dialogue: 0,0:15:25.37,0:15:31.06,Default,,0000,0000,0000,,X minus four, so we have X\Nsquared minus nine times X minus
Dialogue: 0,0:15:31.06,0:15:32.37,Default,,0000,0000,0000,,4 equals 0.
Dialogue: 0,0:15:34.14,0:15:39.76,Default,,0000,0000,0000,,X squared minus nine is the\Ndifference of two squares, so we
Dialogue: 0,0:15:39.76,0:15:45.84,Default,,0000,0000,0000,,can write that in as X plus\Nthree times X minus three times
Dialogue: 0,0:15:45.84,0:15:48.18,Default,,0000,0000,0000,,X minus 4 equals 0.
Dialogue: 0,0:15:48.82,0:15:56.09,Default,,0000,0000,0000,,Giving a solutions X equals\Nminus 3, three or four.
Dialogue: 0,0:15:58.42,0:16:01.06,Default,,0000,0000,0000,,You may have noticed that in\Neach example that we've done.
Dialogue: 0,0:16:02.05,0:16:07.78,Default,,0000,0000,0000,,Every root. There's a factor of\Nthe constant term in the
Dialogue: 0,0:16:07.78,0:16:11.74,Default,,0000,0000,0000,,original equation. For instance,\Nthree and four, both divided
Dialogue: 0,0:16:11.74,0:16:16.44,Default,,0000,0000,0000,,into 36. As long as the\Ncoefficient of X cubed is one,
Dialogue: 0,0:16:16.44,0:16:18.03,Default,,0000,0000,0000,,this must be the case.
Dialogue: 0,0:16:18.78,0:16:25.29,Default,,0000,0000,0000,,Because. The constant is simply\Nthe product of the roots 3 * 3 *
Dialogue: 0,0:16:25.29,0:16:27.20,Default,,0000,0000,0000,,4 is equal to 36.
Dialogue: 0,0:16:27.71,0:16:30.31,Default,,0000,0000,0000,,This gives us another possible
Dialogue: 0,0:16:30.31,0:16:36.98,Default,,0000,0000,0000,,approach. Consider the\Nequation X cubed
Dialogue: 0,0:16:36.98,0:16:40.20,Default,,0000,0000,0000,,minus six X
Dialogue: 0,0:16:40.20,0:16:46.89,Default,,0000,0000,0000,,squared. Minus six X\Nminus 7 equals 0.
Dialogue: 0,0:16:47.84,0:16:50.98,Default,,0000,0000,0000,,Now it's possible for every\Nsolution to be irrational, but
Dialogue: 0,0:16:50.98,0:16:54.43,Default,,0000,0000,0000,,if there is a rational solution,\Nthen because the coefficient of
Dialogue: 0,0:16:54.43,0:16:59.14,Default,,0000,0000,0000,,X cubed is one, it's going to be\Nan integer, and it's going to be
Dialogue: 0,0:16:59.14,0:17:00.40,Default,,0000,0000,0000,,a factor of 7.
Dialogue: 0,0:17:01.01,0:17:05.46,Default,,0000,0000,0000,,This only leaves us with four\Npossibilities, 1 - 1 Seven and
Dialogue: 0,0:17:05.46,0:17:09.54,Default,,0000,0000,0000,,minus Seven, so we can try each\Nof them in turn.
Dialogue: 0,0:17:10.25,0:17:14.84,Default,,0000,0000,0000,,You can see fairly quickly that\None and minus one don't work, so
Dialogue: 0,0:17:14.84,0:17:15.90,Default,,0000,0000,0000,,let's try 7.
Dialogue: 0,0:17:17.00,0:17:20.13,Default,,0000,0000,0000,,Rather than substituting 7 into\Nthis expression and having to
Dialogue: 0,0:17:20.13,0:17:24.20,Default,,0000,0000,0000,,workout 7 cubed and so on, what\NI'm going to do is synthetically
Dialogue: 0,0:17:24.20,0:17:29.21,Default,,0000,0000,0000,,divide by 7, because if Seven is\Na route I'll end up with a 0 and
Dialogue: 0,0:17:29.21,0:17:32.65,Default,,0000,0000,0000,,What's more, the division will\Ngive me the quadratic that I'm
Dialogue: 0,0:17:32.65,0:17:35.90,Default,,0000,0000,0000,,looking for. So let's\Nsynthetically divide this
Dialogue: 0,0:17:35.90,0:17:37.14,Default,,0000,0000,0000,,expression by 7.
Dialogue: 0,0:17:38.79,0:17:40.41,Default,,0000,0000,0000,,The coefficient of X cubed is
Dialogue: 0,0:17:40.41,0:17:43.34,Default,,0000,0000,0000,,one. Coefficient of X squared is
Dialogue: 0,0:17:43.34,0:17:48.61,Default,,0000,0000,0000,,minus 6. The coefficient of X is\Nminus 6 and the constant is
Dialogue: 0,0:17:48.61,0:17:51.53,Default,,0000,0000,0000,,minus Seven and were\Nsynthetically dividing by 7.
Dialogue: 0,0:17:54.60,0:18:00.10,Default,,0000,0000,0000,,Bring the one down 1 * 7 is\NSeven 7 - 6 is one.
Dialogue: 0,0:18:00.95,0:18:04.42,Default,,0000,0000,0000,,1 * 7 is 7 Seven and minus six
Dialogue: 0,0:18:04.42,0:18:07.33,Default,,0000,0000,0000,,is one. 1 * 7 is 7.
Dialogue: 0,0:18:08.13,0:18:10.47,Default,,0000,0000,0000,,7 and minus 70.
Dialogue: 0,0:18:10.97,0:18:16.77,Default,,0000,0000,0000,,So 7 is indeed a route, and\Nthe resulting quadratic is
Dialogue: 0,0:18:16.77,0:18:20.98,Default,,0000,0000,0000,,X squared plus X plus one\Nequals not.
Dialogue: 0,0:18:22.60,0:18:25.91,Default,,0000,0000,0000,,Now you'll find if you try to\Nsolve it that the quadratic
Dialogue: 0,0:18:25.91,0:18:27.57,Default,,0000,0000,0000,,equation X squared plus X plus
Dialogue: 0,0:18:27.57,0:18:31.95,Default,,0000,0000,0000,,one equals not. Has no\Nreal solutions, so the
Dialogue: 0,0:18:31.95,0:18:36.80,Default,,0000,0000,0000,,only possible solution to\Nthis cubic is X equals 7.
Dialogue: 0,0:18:39.52,0:18:44.51,Default,,0000,0000,0000,,Certain basic identity's which\Nyou may wish to learn can
Dialogue: 0,0:18:44.51,0:18:49.00,Default,,0000,0000,0000,,help. In fact, rising both\Ncubics and quadratics. I'll
Dialogue: 0,0:18:49.00,0:18:51.50,Default,,0000,0000,0000,,just give you 1 example.
Dialogue: 0,0:18:52.73,0:18:56.70,Default,,0000,0000,0000,,We have the equation X cubed\Nplus three X squared.
Dialogue: 0,0:18:57.53,0:19:02.27,Default,,0000,0000,0000,,Plus 3X plus\None equals 0.
Dialogue: 0,0:19:04.33,0:19:11.37,Default,,0000,0000,0000,,1331 is the standard expansion\Nof X plus one cubed.
Dialogue: 0,0:19:12.90,0:19:18.27,Default,,0000,0000,0000,,So the solution to this equation\Nis X plus one cubed equals 0.
Dialogue: 0,0:19:18.80,0:19:23.66,Default,,0000,0000,0000,,We have The root minus\NOne X equals minus one
Dialogue: 0,0:19:23.66,0:19:25.07,Default,,0000,0000,0000,,repeated three times.
Dialogue: 0,0:19:27.69,0:19:34.96,Default,,0000,0000,0000,,If you can't find a factor\Nby these methods, then draw an
Dialogue: 0,0:19:34.96,0:19:37.99,Default,,0000,0000,0000,,accurate graph of the cubic
Dialogue: 0,0:19:37.99,0:19:41.96,Default,,0000,0000,0000,,expression. The points where it\Ncrosses the X axis.
Dialogue: 0,0:19:42.67,0:19:44.27,Default,,0000,0000,0000,,Will give you the solutions to
Dialogue: 0,0:19:44.27,0:19:48.50,Default,,0000,0000,0000,,the equation. But their accuracy\Nwill be limited to the accuracy
Dialogue: 0,0:19:48.50,0:19:49.57,Default,,0000,0000,0000,,of your graph.
Dialogue: 0,0:19:50.60,0:19:54.66,Default,,0000,0000,0000,,You might indeed find the graph\Ncrosses the X axis at a point
Dialogue: 0,0:19:54.66,0:19:58.40,Default,,0000,0000,0000,,that would suggest a factor. For\Ninstance, if you draw craft a
Dialogue: 0,0:19:58.40,0:20:01.52,Default,,0000,0000,0000,,graph that appears to cross the\NX axis would say.
Dialogue: 0,0:20:02.08,0:20:06.46,Default,,0000,0000,0000,,X equals 1/2. Then it's worth\Ntrying to find out if X minus
Dialogue: 0,0:20:06.46,0:20:08.15,Default,,0000,0000,0000,,1/2 is indeed a factor.
Dialogue: 0,0:20:10.13,0:20:12.08,Default,,0000,0000,0000,,Let's look at the equation.
Dialogue: 0,0:20:12.59,0:20:14.29,Default,,0000,0000,0000,,X cubed
Dialogue: 0,0:20:15.58,0:20:18.19,Default,,0000,0000,0000,,Plus four X squared.
Dialogue: 0,0:20:19.03,0:20:23.64,Default,,0000,0000,0000,,Plus X minus\N5 equals 0.
Dialogue: 0,0:20:26.90,0:20:30.09,Default,,0000,0000,0000,,Now this equation won't yield\Naffected by any of the methods
Dialogue: 0,0:20:30.09,0:20:32.99,Default,,0000,0000,0000,,that I've discussed, so it's\Ntime to draw a graph.
Dialogue: 0,0:20:36.27,0:20:40.59,Default,,0000,0000,0000,,And this is the graph of Y\Nequals X cubed plus four X
Dialogue: 0,0:20:40.59,0:20:42.25,Default,,0000,0000,0000,,squared plus 6 - 5.
Dialogue: 0,0:20:44.45,0:20:49.48,Default,,0000,0000,0000,,It crosses the X axis at three\Nplaces, 1 near X equals minus 3,
Dialogue: 0,0:20:49.48,0:20:51.63,Default,,0000,0000,0000,,one near X equals minus 2.
Dialogue: 0,0:20:52.16,0:20:54.32,Default,,0000,0000,0000,,And one near X equals 1.
Dialogue: 0,0:20:56.01,0:20:59.72,Default,,0000,0000,0000,,Pending on how accurate your\Ngraph is, you may be able to
Dialogue: 0,0:20:59.72,0:21:03.12,Default,,0000,0000,0000,,pinpoint it a bit closer than\Nthis, and we get solutions.
Dialogue: 0,0:21:03.75,0:21:07.09,Default,,0000,0000,0000,,X is approximately equal to
Dialogue: 0,0:21:07.09,0:21:11.21,Default,,0000,0000,0000,,minus 3.2.\NMinus 1.7
Dialogue: 0,0:21:11.21,0:21:14.41,Default,,0000,0000,0000,,and not .9.
Dialogue: 0,0:21:16.23,0:21:18.95,Default,,0000,0000,0000,,These solutions may be\Naccurate enough for your
Dialogue: 0,0:21:18.95,0:21:22.35,Default,,0000,0000,0000,,needs, but if you require more\Naccurate answers then you
Dialogue: 0,0:21:22.35,0:21:24.73,Default,,0000,0000,0000,,should use a numerical\Nalgorithm using the
Dialogue: 0,0:21:24.73,0:21:28.13,Default,,0000,0000,0000,,approximate answers. If you\Nobtain from the graph as a
Dialogue: 0,0:21:28.13,0:21:28.81,Default,,0000,0000,0000,,starting point.