WEBVTT 00:00:05.000 --> 00:00:14.080 The factors of the constant term must add to the coefficient of the linear term. 00:00:26.480 --> 00:00:30.660 We need two factors, called a factor pair. 00:00:31.640 --> 00:00:36.120 Look for a factor pair of the constant term. 00:00:37.820 --> 00:00:40.580 What must that factor pair do? 00:00:41.140 --> 00:00:43.260 The pair must add to the coefficient of the linear term. 00:00:52.580 --> 00:01:02.220 For this example, use three and four because three plus four is seven. 00:01:04.020 --> 00:01:06.900 Seven is the coefficient of the linear term. 00:01:20.320 --> 00:01:32.400 For this example, the factors are two and six. 00:01:32.960 --> 00:01:42.960 The two numbers multiply to twelve and add to eight. 00:01:42.960 --> 00:01:47.922 For a negative middle term, negative twelve and negative one multiply to twelve and 00:01:47.922 --> 00:01:49.340 add to negative thirteen. 00:02:00.160 --> 00:02:10.980 The first terms of the binomials must multiply to x squared. 00:02:10.980 --> 00:02:20.593 For this example, use negative three and negative four because they add to negative 00:02:20.593 --> 00:02:21.343 seven. 00:03:05.120 --> 00:03:15.540 Both groups now contain the same binomial factor. 00:03:33.360 --> 00:03:35.200 This is a quadratic trinomial. 00:03:40.200 --> 00:03:41.740 It's going to be an x here and an x here. 00:03:52.520 --> 00:04:06.820 Now determine the constant terms in the two binomial factors. 00:04:06.820 --> 00:04:11.280 List the factor pairs of negative twenty-seven. 00:04:44.940 --> 00:04:59.760 The sums are twenty-six, negative twenty-six, six, and negative six. 00:05:06.360 --> 00:05:09.500 The required pair is negative three and positive nine. 00:05:10.640 --> 00:05:17.793 We need factor pairs of the constant term that add to the coefficient of the linear 00:05:17.793 --> 00:05:18.543 term. 00:05:28.300 --> 00:05:32.340 Rewrite six x as negative three x plus nine x, then factor by grouping. 00:05:53.680 --> 00:05:57.660 The result is x plus nine times x minus three. 00:06:27.740 --> 00:06:28.880 The factors are the same product in a different order. 00:06:32.680 --> 00:06:33.600 Multiplication is commutative, so the order does not matter. 00:07:21.580 --> 00:07:29.000 The first term in each binomial is x because x times x is x squared. 00:07:48.820 --> 00:07:51.440 Before listing pairs, use the sign of the linear coefficient. 00:07:53.340 --> 00:07:57.560 The coefficient of the linear term is negative. What does that tell us? 00:08:00.360 --> 00:08:08.420 The constant is positive and the sum is negative, so both factors are negative. 00:08:29.220 --> 00:08:35.960 The pair negative two and negative eight has sum negative ten. 00:08:36.700 --> 00:08:39.760 Therefore the factorization is x minus two times x minus eight. 00:09:02.140 --> 00:09:04.020 Check the factorization by multiplying. 00:09:07.220 --> 00:09:21.040 The product is x squared minus ten x plus sixteen. 00:09:21.760 --> 00:09:27.200 Try another example. 00:09:46.160 --> 00:09:49.900 Find factor pairs of negative sixty that add to negative four. 00:11:30.120 --> 00:11:33.240 A negative constant requires one positive and one negative factor. 00:11:41.600 --> 00:11:44.560 Their values must differ by four. 00:11:50.040 --> 00:11:57.800 Not every polynomial factors over the integers. 00:12:42.420 --> 00:12:45.000 We are going to learn another method later. 00:13:07.900 --> 00:13:12.460 The factor pair is positive seven and negative six. 00:13:28.320 --> 00:13:31.860 We want to recognize these factor pairs quickly. 00:13:43.120 --> 00:13:51.600 This is a quadratic binomial rather than a quadratic trinomial. 00:14:04.620 --> 00:14:12.420 Treat the missing linear coefficient as zero. 00:14:20.180 --> 00:14:25.420 The same method works for a quadratic binomial with no linear term. 00:15:01.850 --> 00:15:07.870 For this quadratic binomial, factor out the greatest common factor first. 00:15:16.930 --> 00:15:27.370 Factoring x gives x times x minus thirty-six. 00:15:57.190 --> 00:16:01.550 Always factor out the greatest common factor before using another method. 00:16:04.280 --> 00:16:07.180 How is this example different? 00:16:10.540 --> 00:16:18.980 The leading coefficient of the quadratic trinomial is not one. 00:16:32.700 --> 00:16:34.020 Possible first terms include four x and x, or two x and two x. 00:17:08.900 --> 00:17:19.620 The constant terms must multiply to positive six. 00:17:44.960 --> 00:17:50.420 Guess-and-check creates too many possibilities, so use a systematic method. 00:18:41.580 --> 00:18:51.720 Four times six is twenty-four. 00:18:57.640 --> 00:19:09.000 Find factor pairs of twenty-four that add to eleven. 00:19:21.560 --> 00:19:22.460 Eight and three multiply to twenty-four and add to eleven. 00:19:25.000 --> 00:19:35.880 Rewrite the middle term as eight x plus three x. 00:19:36.320 --> 00:19:41.240 This changes eleven x to eight x plus three x without changing the expression. 00:19:47.620 --> 00:19:49.100 Now there are four terms, so factor by grouping. 00:20:13.860 --> 00:20:17.980 The result is four x plus three times x plus two. 00:20:30.560 --> 00:20:34.424 Multiply the leading coefficient by the constant, then find a pair with the required 00:20:34.424 --> 00:20:35.174 sum. 00:20:47.980 --> 00:20:56.140 Use the pair to split the middle term, then factor by grouping. 00:21:24.950 --> 00:21:27.530 For this example, find factor pairs of four times negative fifteen. 00:21:32.770 --> 00:21:36.350 The product is negative sixty and the required sum is negative seven. 00:22:10.870 --> 00:22:18.690 The pair five and negative twelve has sum negative seven. 00:22:37.450 --> 00:22:45.990 Rewrite negative seven x as five x minus twelve x, then factor by grouping. 00:23:25.770 --> 00:23:30.810 This method takes more work, but it is systematic. 00:23:41.330 --> 00:23:42.570 The result is correct, and the order of the two binomial factors does not matter. 00:24:00.480 --> 00:24:02.240 This is a cubic trinomial rather than a quadratic trinomial. 00:24:08.920 --> 00:24:11.380 Before using another method, factor out the greatest common factor. 00:24:13.460 --> 00:24:15.480 The greatest common factor is ten x. 00:24:29.760 --> 00:24:36.160 After factoring out ten x, factor the remaining quadratic trinomial. 00:26:05.480 --> 00:26:11.680 Multiply the factors to check the result.