1 00:00:00,000 --> 00:00:00,560 2 00:00:00,560 --> 00:00:04,990 We're told to factor r squared plus 4r, minus 45. 3 00:00:04,990 --> 00:00:07,740 And there are simpler ways to factor this, because this has 4 00:00:07,740 --> 00:00:11,160 1 as a leading coefficient on the r squared right here, but 5 00:00:11,160 --> 00:00:11,990 what we're going to do is we're going to 6 00:00:11,990 --> 00:00:14,419 factor this by grouping. 7 00:00:14,419 --> 00:00:16,750 And when you factor it by grouping, what you want to do 8 00:00:16,750 --> 00:00:20,109 is find two numbers whose product-- so the two numbers 9 00:00:20,109 --> 00:00:22,919 are a and b-- whose product is equal to the coefficient on r 10 00:00:22,920 --> 00:00:27,910 squared, 1 times negative 45, which is just negative 45. 11 00:00:27,910 --> 00:00:30,899 And whose sum is equal to the coefficient on the first 12 00:00:30,899 --> 00:00:33,449 degree term, is equal to 4. 13 00:00:33,450 --> 00:00:37,130 a plus b is equal to 4. 14 00:00:37,130 --> 00:00:38,950 So let's look at all of the different combinations that 15 00:00:38,950 --> 00:00:41,179 goes to negative 45, and see if when you add them, 16 00:00:41,179 --> 00:00:43,089 you ever get to 4. 17 00:00:43,090 --> 00:00:44,395 So let's take the combinations. 18 00:00:44,395 --> 00:00:46,980 19 00:00:46,979 --> 00:00:48,909 Let's look at their different a's and b's, and let's see 20 00:00:48,909 --> 00:00:51,119 what happens when you sum them. 21 00:00:51,119 --> 00:00:55,199 So I could try 1 and negative 45, but if you take their sum, 22 00:00:55,200 --> 00:00:57,260 that's not going to get you anywhere close to 4, even if 23 00:00:57,259 --> 00:01:00,109 you swapped them, negative 1 and positive 45. 24 00:01:00,109 --> 00:01:05,030 Let's see, if you do 3 and 15, 3 and negative 15, or negative 25 00:01:05,030 --> 00:01:07,150 3 and 15, those are way more than 4 apart. 26 00:01:07,150 --> 00:01:07,890 We're going to have to find numbers 27 00:01:07,890 --> 00:01:09,930 that are closer together. 28 00:01:09,930 --> 00:01:13,660 So let's see, 3, 4, 5 and 9, that looks interesting. 29 00:01:13,659 --> 00:01:16,119 5 and 9 are 4 apart. 30 00:01:16,120 --> 00:01:24,040 And if we go with 5 and negative 9, what's their sum? 31 00:01:24,040 --> 00:01:25,930 Their sum is negative 4. 32 00:01:25,930 --> 00:01:30,120 What happens if we go with negative 5 and positive 9? 33 00:01:30,120 --> 00:01:33,120 Well, then their sum is 4, so negative 5 34 00:01:33,120 --> 00:01:35,560 and positive 9 works. 35 00:01:35,560 --> 00:01:38,460 So what we want to do here, now that we found the numbers 36 00:01:38,459 --> 00:01:41,349 whose product is equal to the 1 times negative 45, and whose 37 00:01:41,349 --> 00:01:46,549 sum is equal to 4, let's break this 4r up into a sum of 38 00:01:46,549 --> 00:01:50,509 negative 5-- let me break it up this way-- 39 00:01:50,510 --> 00:01:55,390 into 9r minus 5r. 40 00:01:55,390 --> 00:01:59,040 I broke that 4r into that using our two numbers, and 41 00:01:59,040 --> 00:02:03,430 then I could put the r squared out front, r squared, and then 42 00:02:03,430 --> 00:02:04,810 we have a minus 45. 43 00:02:04,810 --> 00:02:07,370 And now we're ready to group this and 44 00:02:07,370 --> 00:02:08,770 then factor by grouping. 45 00:02:08,770 --> 00:02:11,840 So let's look at this first group right here. 46 00:02:11,840 --> 00:02:15,090 They both have the common factor of r, so let's 47 00:02:15,090 --> 00:02:16,159 factor out an r. 48 00:02:16,159 --> 00:02:19,900 This is equal to r times-- r square divided by r is r. 49 00:02:19,900 --> 00:02:22,890 9r divided by r is 9. 50 00:02:22,889 --> 00:02:24,699 So that's that first group. 51 00:02:24,699 --> 00:02:29,509 And the second group right here, has a common factor of 52 00:02:29,509 --> 00:02:30,509 negative 5. 53 00:02:30,509 --> 00:02:31,759 So let's factor it out. 54 00:02:31,759 --> 00:02:34,469 55 00:02:34,469 --> 00:02:37,210 Negative 5r divided by negative 5 is just an r. 56 00:02:37,210 --> 00:02:41,240 Negative 45 divided by negative 5 is a 9. 57 00:02:41,240 --> 00:02:45,820 And then we have an r plus 9 in both terms. r times r plus 58 00:02:45,819 --> 00:02:49,060 9, negative 5 times r plus 9. 59 00:02:49,060 --> 00:02:50,710 So we can factor that out. 60 00:02:50,710 --> 00:02:58,290 So you factor an r plus 9 out, and you're left with r times r 61 00:02:58,289 --> 00:03:02,039 plus 9, divided by r plus 9 is just going to be an r. 62 00:03:02,039 --> 00:03:07,349 And then negative 5 times r plus 9, divided by r plus 9 is 63 00:03:07,349 --> 00:03:10,199 negative 5, and we're done!