1 00:00:05,000 --> 00:00:14,080 The factors of the constant term must add to the coefficient of the linear term. 2 00:00:26,480 --> 00:00:30,660 We need two factors, called a factor pair. 3 00:00:31,640 --> 00:00:36,120 Look for a factor pair of the constant term. 4 00:00:37,820 --> 00:00:40,580 What must that factor pair do? 5 00:00:41,140 --> 00:00:43,260 The pair must add to the coefficient of the linear term. 6 00:00:52,580 --> 00:01:02,220 For this example, use three and four because three plus four is seven. 7 00:01:04,020 --> 00:01:06,900 Seven is the coefficient of the linear term. 8 00:01:20,320 --> 00:01:32,400 For this example, the factors are two and six. 9 00:01:32,960 --> 00:01:42,960 The two numbers multiply to twelve and add to eight. 10 00:01:42,960 --> 00:01:47,922 For a negative middle term, negative twelve and negative one multiply to twelve and 11 00:01:47,922 --> 00:01:49,340 add to negative thirteen. 12 00:02:00,160 --> 00:02:10,980 The first terms of the binomials must multiply to x squared. 13 00:02:10,980 --> 00:02:20,593 For this example, use negative three and negative four because they add to negative 14 00:02:20,593 --> 00:02:21,343 seven. 15 00:03:05,120 --> 00:03:15,540 Both groups now contain the same binomial factor. 16 00:03:33,360 --> 00:03:35,200 This is a quadratic trinomial. 17 00:03:40,200 --> 00:03:41,740 It's going to be an x here and an x here. 18 00:03:52,520 --> 00:04:06,820 Now determine the constant terms in the two binomial factors. 19 00:04:06,820 --> 00:04:11,280 List the factor pairs of negative twenty-seven. 20 00:04:44,940 --> 00:04:59,760 The sums are twenty-six, negative twenty-six, six, and negative six. 21 00:05:06,360 --> 00:05:09,500 The required pair is negative three and positive nine. 22 00:05:10,640 --> 00:05:17,793 We need factor pairs of the constant term that add to the coefficient of the linear 23 00:05:17,793 --> 00:05:18,543 term. 24 00:05:28,300 --> 00:05:32,340 Rewrite six x as negative three x plus nine x, then factor by grouping. 25 00:05:53,680 --> 00:05:57,660 The result is x plus nine times x minus three. 26 00:06:27,740 --> 00:06:28,880 The factors are the same product in a different order. 27 00:06:32,680 --> 00:06:33,600 Multiplication is commutative, so the order does not matter. 28 00:07:21,580 --> 00:07:29,000 The first term in each binomial is x because x times x is x squared. 29 00:07:48,820 --> 00:07:51,440 Before listing pairs, use the sign of the linear coefficient. 30 00:07:53,340 --> 00:07:57,560 The coefficient of the linear term is negative. What does that tell us? 31 00:08:00,360 --> 00:08:08,420 The constant is positive and the sum is negative, so both factors are negative. 32 00:08:29,220 --> 00:08:35,960 The pair negative two and negative eight has sum negative ten. 33 00:08:36,700 --> 00:08:39,760 Therefore the factorization is x minus two times x minus eight. 34 00:09:02,140 --> 00:09:04,020 Check the factorization by multiplying. 35 00:09:07,220 --> 00:09:21,040 The product is x squared minus ten x plus sixteen. 36 00:09:21,760 --> 00:09:27,200 Try another example. 37 00:09:46,160 --> 00:09:49,900 Find factor pairs of negative sixty that add to negative four. 38 00:11:30,120 --> 00:11:33,240 A negative constant requires one positive and one negative factor. 39 00:11:41,600 --> 00:11:44,560 Their values must differ by four. 40 00:11:50,040 --> 00:11:57,800 Not every polynomial factors over the integers. 41 00:12:42,420 --> 00:12:45,000 We are going to learn another method later. 42 00:13:07,900 --> 00:13:12,460 The factor pair is positive seven and negative six. 43 00:13:28,320 --> 00:13:31,860 We want to recognize these factor pairs quickly. 44 00:13:43,120 --> 00:13:51,600 This is a quadratic binomial rather than a quadratic trinomial. 45 00:14:04,620 --> 00:14:12,420 Treat the missing linear coefficient as zero. 46 00:14:20,180 --> 00:14:25,420 The same method works for a quadratic binomial with no linear term. 47 00:15:01,850 --> 00:15:07,870 For this quadratic binomial, factor out the greatest common factor first. 48 00:15:16,930 --> 00:15:27,370 Factoring x gives x times x minus thirty-six. 49 00:15:57,190 --> 00:16:01,550 Always factor out the greatest common factor before using another method. 50 00:16:04,280 --> 00:16:07,180 How is this example different? 51 00:16:10,540 --> 00:16:18,980 The leading coefficient of the quadratic trinomial is not one. 52 00:16:32,700 --> 00:16:34,020 Possible first terms include four x and x, or two x and two x. 53 00:17:08,900 --> 00:17:19,620 The constant terms must multiply to positive six. 54 00:17:44,960 --> 00:17:50,420 Guess-and-check creates too many possibilities, so use a systematic method. 55 00:18:41,580 --> 00:18:51,720 Four times six is twenty-four. 56 00:18:57,640 --> 00:19:09,000 Find factor pairs of twenty-four that add to eleven. 57 00:19:21,560 --> 00:19:22,460 Eight and three multiply to twenty-four and add to eleven. 58 00:19:25,000 --> 00:19:35,880 Rewrite the middle term as eight x plus three x. 59 00:19:36,320 --> 00:19:41,240 This changes eleven x to eight x plus three x without changing the expression. 60 00:19:47,620 --> 00:19:49,100 Now there are four terms, so factor by grouping. 61 00:20:13,860 --> 00:20:17,980 The result is four x plus three times x plus two. 62 00:20:30,560 --> 00:20:34,424 Multiply the leading coefficient by the constant, then find a pair with the required 63 00:20:34,424 --> 00:20:35,174 sum. 64 00:20:47,980 --> 00:20:56,140 Use the pair to split the middle term, then factor by grouping. 65 00:21:24,950 --> 00:21:27,530 For this example, find factor pairs of four times negative fifteen. 66 00:21:32,770 --> 00:21:36,350 The product is negative sixty and the required sum is negative seven. 67 00:22:10,870 --> 00:22:18,690 The pair five and negative twelve has sum negative seven. 68 00:22:37,450 --> 00:22:45,990 Rewrite negative seven x as five x minus twelve x, then factor by grouping. 69 00:23:25,770 --> 00:23:30,810 This method takes more work, but it is systematic. 70 00:23:41,330 --> 00:23:42,570 The result is correct, and the order of the two binomial factors does not matter. 71 00:24:00,480 --> 00:24:02,240 This is a cubic trinomial rather than a quadratic trinomial. 72 00:24:08,920 --> 00:24:11,380 Before using another method, factor out the greatest common factor. 73 00:24:13,460 --> 00:24:15,480 The greatest common factor is ten x. 74 00:24:29,760 --> 00:24:36,160 After factoring out ten x, factor the remaining quadratic trinomial. 75 00:26:05,480 --> 00:26:11,680 Multiply the factors to check the result.