WEBVTT Kind: captions Language: en 00:00:01.380 --> 00:00:04.760 >> Welcome to the Cypress College Math Review 00:00:05.376 --> 00:00:08.006 on Partial Fraction Decomposition. 00:00:12.836 --> 00:00:17.006 Partial fraction decomposition is a method of breaking 00:00:17.006 --> 00:00:20.316 up a complicated rational expression 00:00:20.806 --> 00:00:23.236 into the sum of simpler ones. 00:00:24.016 --> 00:00:28.186 First factor the denominator of the rational expression. 00:00:29.316 --> 00:00:33.756 Based on the chart below determine what terms will be 00:00:33.756 --> 00:00:34.946 in your decomposition. 00:00:37.476 --> 00:00:41.976 When you factor your denominator if you end up with a factor 00:00:42.266 --> 00:00:46.746 that looks like this, that's the first power of a linear, 00:00:47.116 --> 00:00:53.056 linear mx + b. So 3x + 1 is the first power of a linear. 00:00:53.806 --> 00:00:59.346 You then will end up with the term a over 3x + 1 00:00:59.426 --> 00:01:00.706 in your decomposition. 00:01:01.656 --> 00:01:07.216 If instead you ended up with the square of 3x + 1, 00:01:07.536 --> 00:01:09.416 that's the second power of a linear. 00:01:09.766 --> 00:01:18.026 Then you would have a over 3x + 1 + b over the square of 3x + 1. 00:01:18.026 --> 00:01:20.376 You would have 2 terms in your decomposition 00:01:20.876 --> 00:01:23.276 because it's the second power of a linear. 00:01:23.906 --> 00:01:36.906 Other examples of first power of linears are and, other examples 00:01:37.220 --> 00:01:42.820 of second power of linear are and. 00:01:44.300 --> 00:01:47.080 Now let's look at an irreducible quadratic. 00:01:47.596 --> 00:01:51.806 An irreducible quadratic is one with no real 0s. 00:01:52.296 --> 00:01:54.976 For example, x squared + 4. 00:01:55.346 --> 00:01:59.606 If you set that equal to 0, you don't get a real number. 00:02:01.146 --> 00:02:07.106 If that is a factor of your denominator, then instead 00:02:07.106 --> 00:02:10.766 of putting a over that in your decomposition, 00:02:11.166 --> 00:02:14.606 you will put ax + b over that. 00:02:15.156 --> 00:02:21.486 If you have the second power of that, then you will put ax + b 00:02:21.486 --> 00:02:24.586 over it and then you will also have a second term 00:02:24.586 --> 00:02:28.996 where you will put cx + d over that squared. 00:02:29.616 --> 00:02:32.856 This is how you set up your decompositions. 00:02:36.886 --> 00:02:40.000 So the first step is to factor the denominator. 00:02:48.560 --> 00:02:52.460 Each of these factors are the first power of a linear. 00:02:54.120 --> 00:03:00.760 So we end up with a over x + 2 and b over x - 1. 00:03:02.520 --> 00:03:06.320 The next step is to multiply by the original denominator. 00:03:09.960 --> 00:03:14.100 Over on the right hand side when I multiply by the denominator, 00:03:14.800 --> 00:03:17.736 it looks like I am cross multiplying. 00:03:23.920 --> 00:03:25.680 There are 2 different methods 00:03:25.846 --> 00:03:27.826 for finishing a problem at this point. 00:03:29.136 --> 00:03:41.006 You can either equate coefficients or substitute 00:03:41.006 --> 00:03:44.126 in values for x. On the first problem, 00:03:44.196 --> 00:03:45.986 I'm going to equate coefficients. 00:03:46.316 --> 00:03:50.666 On the other problems, I will substitute in values for x. 00:03:52.180 --> 00:03:56.060 So to equate coefficients, I distribute 00:04:05.460 --> 00:04:08.700 and basically I am adding like terms 00:04:09.460 --> 00:04:13.120 and factoring out the powers of x. 00:04:28.560 --> 00:04:37.660 So on the left, I have 3x, on the right I have (a + b)x. 00:04:38.656 --> 00:04:42.316 So this is supposed to be true for all x, therefore, 00:04:42.316 --> 00:04:44.976 3 is supposed to equal a+b. 00:04:46.100 --> 00:04:51.136 On the left hand side, my constant is 0, 00:04:52.080 --> 00:04:56.600 on the right hand side, my constant is negative a + 2b. 00:04:57.540 --> 00:05:02.320 Therefore, 0 is equal to negative a + 2b 00:05:02.840 --> 00:05:07.880 and then you solve the system in any way you would like to. 00:05:18.500 --> 00:05:21.040 So b is equal to 1. 00:05:26.460 --> 00:05:28.680 So a is equal to 2. 00:05:29.136 --> 00:05:36.526 So the answer to the problem is 2 over x + 2 + 1 over x - 1. 00:05:43.316 --> 00:05:47.806 Before breaking up a rational expression using partial 00:05:47.806 --> 00:05:52.256 fraction decomposition you must first make sure the expression 00:05:52.256 --> 00:05:52.996 is proper. 00:05:53.866 --> 00:05:57.226 A proper rational expression is one in which the degree 00:05:57.226 --> 00:06:01.306 of the numerator is less than the degree of the denominator. 00:06:02.386 --> 00:06:06.886 If the rational expression is improper, then divide first. 00:06:07.966 --> 00:06:11.066 Use partial fraction decomposition to break 00:06:11.066 --> 00:06:13.720 up the rational expression that you're left with. 00:06:17.880 --> 00:06:22.000 On this problem, you notice that the degree of the numerator is 3 00:06:22.866 --> 00:06:25.116 and the degree of the denominator is 2. 00:06:25.706 --> 00:06:26.986 So we have to divide. 00:06:28.140 --> 00:06:31.040 In fact if the degrees were the same, 00:06:31.240 --> 00:06:32.880 you would also have to divide. 00:06:45.220 --> 00:06:46.700 So we do our long division. 00:06:53.856 --> 00:07:01.816 We subtract we get negative 2x squared + 4x - 3. 00:07:03.276 --> 00:07:05.376 That goes in negative 2 times 00:07:06.186 --> 00:07:12.666 so we get negative 2x squared - 6x + 8. 00:07:13.696 --> 00:07:18.056 We subtract and get 10x - 11. 00:07:18.936 --> 00:07:25.816 So, our quotient is x - 2, our remainder is 10x - 00:07:26.186 --> 00:07:33.616 11 and our divisor is x squared + 3x -4. 00:07:35.266 --> 00:07:38.996 We still need to do a partial fraction decomposition 00:07:39.346 --> 00:07:40.866 on this fraction here. 00:07:42.066 --> 00:07:44.436 So eventually we'll put our answer down here. 00:07:45.060 --> 00:07:50.760 So let's factor our denominator. 00:07:58.080 --> 00:07:59.880 Both first power of linear. 00:08:00.400 --> 00:08:10.600 Multiply by the denominator, the original denominator. 00:08:12.280 --> 00:08:14.160 It looks like we're cross multiplying. 00:08:19.760 --> 00:08:21.360 This time I'm going to substitute 00:08:21.360 --> 00:08:25.440 in our critical values of 1 and negative 4. 00:08:30.760 --> 00:08:32.860 That will make this term disappear. 00:08:33.316 --> 00:08:39.816 So if we let x be 1 on the left hand side, we get this. 00:08:40.736 --> 00:08:42.106 On the right hand side, 00:08:42.106 --> 00:08:46.056 the first term disappears and we get 5b. 00:08:46.626 --> 00:08:51.156 So we get negative 1 is equal to 5b. 00:08:52.826 --> 00:08:56.036 So b is equal to negative 1/5. 00:08:57.720 --> 00:09:04.580 Now we'll let x be the other critical value, negative 4. 00:09:06.280 --> 00:09:08.540 This time this term will disappear 00:09:15.000 --> 00:09:19.620 and we'll have negative 5a on the right hand side. 00:09:20.186 --> 00:09:29.806 So a will be 51/5. 00:09:31.666 --> 00:09:36.486 So the partial fraction decomposition gives us a result 00:09:36.746 --> 00:09:47.706 of 51/5 over x + 4 - 1/5 over x - 1, 00:09:48.706 --> 00:09:51.726 but normally you don't write fractions inside fractions; 00:09:51.726 --> 00:09:53.206 that's a complex fraction. 00:09:54.086 --> 00:09:57.646 So let's multiply 5 over 5 to those fractions 00:09:57.906 --> 00:10:01.316 as we put our answer over on the left hand side here 00:10:01.746 --> 00:10:03.126 with our X -2. 00:10:03.796 --> 00:10:10.606 So we have 51 on the top and we have 5 times the quantity X + 4 00:10:11.176 --> 00:10:20.126 and then we'll change that to subtraction and then 1 00:10:20.556 --> 00:10:26.796 over 5 times the quantity x - 1 and there's our answer. 00:10:31.276 --> 00:10:37.056 All right here we go it's all factored for us and ready to go. 00:10:37.296 --> 00:10:39.276 So we have a over x + 1. 00:10:41.156 --> 00:10:44.316 The second factor is an irreducible quadratic. 00:10:45.026 --> 00:10:50.556 So instead of just b over that factor we will have bx + c 00:10:51.900 --> 00:10:54.320 multiply by the original denominator. 00:11:00.260 --> 00:11:02.060 Watch your parentheses over here. 00:11:12.320 --> 00:11:14.826 So let's let x be negative 1. 00:11:14.826 --> 00:11:27.556 So on the left hand side, we get 1 + 4 + 7. 00:11:28.106 --> 00:11:31.196 On the right hand side, this term goes away 00:11:31.746 --> 00:11:38.896 but over here we have a times the quantity 1 + 2 + 3 00:11:40.080 --> 00:11:45.940 which gives me 12 is equal to 6a so a is 2. 00:11:49.520 --> 00:11:52.260 And now we don't have any more critical values 00:11:52.606 --> 00:11:54.946 so we just pick numbers we like. 00:11:55.440 --> 00:11:57.836 So let x be 0. 00:11:59.260 --> 00:12:02.320 So on the left hand side I got 7, 00:12:02.866 --> 00:12:11.596 on the right hand side I got 3a + c. I already know that a is 2, 00:12:12.956 --> 00:12:20.416 which gave me that c is 1 and then let's let x be say 1. 00:12:21.196 --> 00:12:28.276 So on the left hand side, I get 1 - 4 + 7 that's equal 00:12:28.276 --> 00:12:39.086 to a times 1 - 2 + 3 + the quantity b + c times 2. 00:12:39.536 --> 00:12:44.106 You'll notice that I'm substituting in for x but not 00:12:44.106 --> 00:12:49.246 for a, b and c. I usually do it for x and then I do it for a, 00:12:49.246 --> 00:12:52.356 b and c on the next step, but whatever works for you. 00:12:53.006 --> 00:13:01.836 So on the left hand side, I get 4 is equal to 4 + 2b + 2 00:13:03.016 --> 00:13:07.766 so negative 2 is equal to 2b which gives us 00:13:07.766 --> 00:13:10.546 that b is equal to negative 1. 00:13:12.026 --> 00:13:17.296 Answer a, which is 2 over x +1, 00:13:18.276 --> 00:13:20.816 +the quantity let's see what have we go? 00:13:21.306 --> 00:13:24.466 Negative 1 for b so that's our coefficient 00:13:24.466 --> 00:13:27.316 for x and then c is 1. 00:13:27.696 --> 00:13:31.916 So negative x + 1 is the numerator in the second fraction 00:13:32.386 --> 00:13:37.116 and our denominator was x squared - 2x + 3. 00:13:45.046 --> 00:13:48.516 The first factor is the first power of a linear. 00:13:51.206 --> 00:13:55.056 The second is the second power of a linear. 00:13:55.546 --> 00:14:02.026 So we not only have b over x + 1, but we also have c 00:14:02.366 --> 00:14:04.926 over the square of x + 1. 00:14:05.476 --> 00:14:12.306 It is not cx + d. Now multiply by the original denominator 00:14:13.496 --> 00:14:17.116 and we end up with x - 3 on the left hand side. 00:14:17.306 --> 00:14:21.946 On the right hand side, we have a times the square 00:14:21.946 --> 00:14:33.836 of x + 1 + b times x + 2 x + 1 + c times x + 2. 00:14:34.820 --> 00:14:37.760 First we'll let x be negative 1. 00:14:39.160 --> 00:14:44.376 In this case, we get negative 4 on the left, on the right, 00:14:44.376 --> 00:14:47.506 we simply get c. So we have our first value. 00:14:49.420 --> 00:14:52.840 Next let x be negative 2, 00:14:53.820 --> 00:14:56.160 on the left hand side we get negative 5, 00:14:56.600 --> 00:15:01.820 on the right hand side we simply get a. So a is negative 5. 00:15:03.560 --> 00:15:06.556 Finally, we run out of critical values 00:15:06.556 --> 00:15:09.496 so we'll let x be some other value say 0. 00:15:09.496 --> 00:15:13.836 On the left hand side, we get negative 3, 00:15:14.966 --> 00:15:17.376 on the right hand side substituting in 0 00:15:17.376 --> 00:15:23.026 for x we get a + 2b + 2c. 00:15:23.476 --> 00:15:25.736 Now substituting in the values we know for a 00:15:25.736 --> 00:15:34.246 and c we have negative 5 + 2b -8. 00:15:34.886 --> 00:15:41.496 So we have 10 is equal to 2b which tells us that b is 5. 00:15:42.036 --> 00:15:49.416 That gives us our answer of negative 5 over x + 2 + 5 00:15:49.856 --> 00:15:57.000 over x + 1 - 4 over the square of x + 1. 00:16:04.356 --> 00:16:08.676 Here's some more problems that you should practice on your own. 00:16:09.426 --> 00:16:11.566 After you have done the problems, 00:16:11.886 --> 00:16:13.396 check with the answers below. 00:16:21.806 --> 00:16:23.976 Here are the answers to the practice problems.