WEBVTT 00:00:00.530 --> 00:00:04.020 - So we're given a p of x, it's a third degree polynomial, 00:00:04.020 --> 00:00:07.120 and they say, plot all the zeroes or the x-intercepts 00:00:07.120 --> 00:00:09.310 of the polynomial in the interactive graph. 00:00:09.310 --> 00:00:10.510 And the reason why they say interactive graph, 00:00:10.510 --> 00:00:13.810 this is a screen shot from the exercise on Kahn Academy, 00:00:13.810 --> 00:00:16.810 where you could click and place the zeroes. 00:00:16.810 --> 00:00:20.340 But the key here is, lets figure out what x values make 00:00:20.340 --> 00:00:22.640 p of x equal to zero, 00:00:22.640 --> 00:00:23.960 those are the zeroes. 00:00:23.960 --> 00:00:25.230 And then we can plot them. 00:00:25.230 --> 00:00:26.063 So pause this video, 00:00:26.063 --> 00:00:28.380 and see if you can figure that out. 00:00:28.380 --> 00:00:32.660 So the key here is to try to factor this expression 00:00:32.660 --> 00:00:34.320 right over here, this third degree expression, 00:00:34.320 --> 00:00:36.850 because really we're trying to solve the X's 00:00:36.850 --> 00:00:40.490 for which five x to third plus five x squared 00:00:40.490 --> 00:00:43.070 minus 30 x is equal to zero. 00:00:43.070 --> 00:00:44.640 And the way we do that is 00:00:44.640 --> 00:00:47.210 by factoring this left-hand expression. 00:00:47.210 --> 00:00:48.890 So the first thing I always look for 00:00:48.890 --> 00:00:51.140 is a common factor across all of the terms. 00:00:51.140 --> 00:00:55.240 It looks like all of the terms are divisible by five x. 00:00:55.240 --> 00:00:57.170 So let's factor out a five x. 00:00:57.170 --> 00:00:58.980 So this is going to be five x times, 00:00:58.980 --> 00:01:01.370 if we take a five x out of five x to the third, 00:01:01.370 --> 00:01:03.040 we're left with an x squared. 00:01:03.040 --> 00:01:04.840 If we take out a five x out of five x squared, 00:01:04.840 --> 00:01:06.800 we're left with an x, so plus x. 00:01:06.800 --> 00:01:09.200 And if we take out a five x of negative 30 x, 00:01:09.200 --> 00:01:13.230 we're left with a negative six is equal to zero. 00:01:13.230 --> 00:01:17.800 And now, we have five x times this second degree, 00:01:17.800 --> 00:01:20.580 the second degree expression and to factor that, 00:01:20.580 --> 00:01:23.230 let's see, what two numbers add up to one? 00:01:23.230 --> 00:01:25.110 You could use as a one x here. 00:01:25.110 --> 00:01:27.730 And their product is equal to negative six. 00:01:27.730 --> 00:01:31.290 And let's see, positive three and negative two 00:01:31.290 --> 00:01:32.550 would do the trick. 00:01:32.550 --> 00:01:35.400 So I can rewrite this as five x times, 00:01:35.400 --> 00:01:39.510 so x plus three, x plus three, 00:01:39.510 --> 00:01:43.410 times x minus two, and if what I did looks unfamiliar, 00:01:43.410 --> 00:01:46.100 I encourage you to review factoring quadratics 00:01:46.100 --> 00:01:47.320 on Kahn Academy, 00:01:47.320 --> 00:01:50.460 and that is all going to be equal to zero. 00:01:50.460 --> 00:01:53.170 And so if I try to figure out what x values 00:01:53.170 --> 00:01:54.780 are going to make this whole expression zero, 00:01:54.780 --> 00:01:56.185 it could be the x values 00:01:56.185 --> 00:01:58.910 or the x value that makes five x equal zero. 00:01:58.910 --> 00:02:00.210 Because if five x zero, 00:02:00.210 --> 00:02:02.980 zero times anything else is going to be zero. 00:02:02.980 --> 00:02:06.000 So what makes five x equal zero? 00:02:06.000 --> 00:02:09.550 Well if we divide five, if you divide both sides by five, 00:02:09.550 --> 00:02:11.340 you're going to get x is equal to zero. 00:02:11.340 --> 00:02:12.450 And it is the case. 00:02:12.450 --> 00:02:15.230 If x equals zero, this becomes zero, 00:02:15.230 --> 00:02:17.240 and then doesn't matter what these are, 00:02:17.240 --> 00:02:18.976 zero times anything is zero. 00:02:18.976 --> 00:02:22.050 The other possible x value that would make everything zero 00:02:22.050 --> 00:02:25.460 is the x value that makes x plus three equal to zero. 00:02:25.460 --> 00:02:26.920 Subtract three from both sides 00:02:26.920 --> 00:02:29.000 you get x is equal to negative three. 00:02:29.000 --> 00:02:31.420 And then the other x value is the x value that makes 00:02:31.420 --> 00:02:34.050 x minus two equal to zero. 00:02:34.050 --> 00:02:37.210 Add two to both sides, that's gonna be x equals two. 00:02:37.210 --> 00:02:38.043 So there you have it. 00:02:38.043 --> 00:02:41.150 We have identified three x values that make our polynomial 00:02:41.150 --> 00:02:43.250 equal to zero and those are going to be the zeros 00:02:43.250 --> 00:02:44.770 and the x intercepts. 00:02:44.770 --> 00:02:47.830 So we have one at x equals zero. 00:02:47.830 --> 00:02:52.260 We have one at x equals negative three. 00:02:52.260 --> 00:02:57.260 We have one at x equals, at x equals two. 00:02:58.620 --> 00:02:59.830 And the reason why it's, 00:02:59.830 --> 00:03:01.370 we're done now with this exercise, 00:03:01.370 --> 00:03:02.580 if you're doing this on Kahn Academy 00:03:02.580 --> 00:03:04.460 or just clicked in these three places, 00:03:04.460 --> 00:03:07.201 but the reason why folks find this to be useful is 00:03:07.201 --> 00:03:10.320 it helps us start to think about what the graph could be. 00:03:10.320 --> 00:03:12.140 Because the graph has to intercept 00:03:12.140 --> 00:03:13.904 the x axis at these points. 00:03:13.904 --> 00:03:18.360 So the graph might look something like that, 00:03:18.360 --> 00:03:21.680 it might look something like that. 00:03:21.680 --> 00:03:24.020 And to figure out what it actually does look like 00:03:24.020 --> 00:03:26.330 we'd probably want to try out a few more x values 00:03:26.330 --> 00:03:28.300 in between these x intercepts 00:03:28.300 --> 00:03:31.083 to get the general sense of the graph.