WEBVTT 00:22.160 --> 00:29.163 In this video, we're going to discuss greatest common factors and factoring by grouping. 00:29.200 --> 00:41.308 In the product two times three which is equal to six, two and three are known as the factors 00:41.360 --> 00:54.655 of six. Two is a factor, and three is a factor of six. Similarly, in algebraic expressions, 00:54.720 --> 01:03.697 x plus two times x plus three is an expression written in factored form. When multiplied together, 01:03.760 --> 01:13.707 they give us the polynomial x squared plus five x plus six. So x plus two and x plus three 01:13.760 --> 01:24.919 are the factors of the polynomial, and again, this is known as factored form. 01:24.960 --> 01:32.827 The greatest common factor, or GCF, of a list of terms or monomials is the product of the 01:32.880 --> 01:39.967 GCF of the numerical coefficients and each GCF of the variable factors. 01:40.000 --> 01:51.245 Let's look at an example. I have two monomials here, twelve m n to the fifth and four m squared 01:51.280 --> 02:00.020 n to the seventh. To find the GCF of each of these two expressions, we begin by looking at the GCF 02:00.080 --> 02:08.762 of the numerical coefficients. The common factors would be the factors that these two coefficients 02:08.800 --> 02:19.073 both have. So let's write out the factors first of twelve. The factors of twelve are one, two, 02:19.120 --> 02:30.751 three, four, six, and twelve. Where the factors of four are one, two, 02:30.800 --> 02:38.359 and four. We can see that they have in common factors of one, two, 02:38.400 --> 02:49.470 and four. So the GCF, or greatest common factor, of the numerical coefficients is four. Now looking 02:49.520 --> 03:01.949 at the variable factors, we have m to the first power and m squared. m and m squared, which is 03:02.000 --> 03:14.028 the same as m times m. So the common factors here are m. They both share an m to the first power. 03:14.080 --> 03:22.736 For the next variable factor, we have n to the fifth and n to the seventh. 03:22.800 --> 03:30.344 n to the fifth can be written as n times itself five times, and n to the seventh 03:30.400 --> 03:37.551 can be written as n times itself seven times. Or another way it can be written 03:37.600 --> 03:52.499 as n to the fifth times n squared. So the greatest common factor of these is n to the fifth. 03:52.560 --> 04:14.755 Multiplying these GCF's together gives us four m n to the fifth. So the GCF is four m n to the fifth. 04:14.800 --> 04:22.196 In these next examples, we are asked to factor each polynomial by factoring out the GCF. 04:22.240 --> 04:28.736 Factoring a polynomial is the reverse process of using the distributive property to multiply 04:28.800 --> 04:38.512 factors together. So we'll begin with fourteen t plus twenty-one. We need to identify the GCF 04:38.560 --> 04:45.719 of the two terms, fourteen t and twenty-one. So recall to do that we look for the product 04:45.760 --> 04:53.393 of the GCF of the numerical coefficients, which would be fourteen and twenty-one, and we multiply 04:53.440 --> 05:08.442 that times the GCF of the variable factors. The GCF of fourteen and twenty-one is seven. 05:08.480 --> 05:15.549 And since there's only a t in this first term and not in the second, there is not a greatest common 05:15.600 --> 05:24.358 factor or common factor at all of the variable factor. So the GCF of the monomials here is seven. 05:24.400 --> 05:32.366 Using the reverse distributive property, we can rewrite this as seven open parentheses 05:32.400 --> 05:39.806 and then divide each of these two terms by the GCF. So fourteen t divided by seven 05:39.840 --> 05:49.616 leaves us with two t, and twenty-one divided by seven leaves us with three, so plus three. This 05:49.680 --> 05:57.391 is factored form of this polynomial. We can check our results by using the distributive property 05:57.440 --> 06:03.463 to distribute seven into the grouping symbols, and you would see that this expression is 06:03.520 --> 06:11.705 equivalent to fourteen t plus twenty-one. Now let's look at our next example. 06:11.738 --> 06:20.280 We have negative six x squared plus three x. Let's begin by finding the GCF of the 06:20.320 --> 06:28.121 two terms. We begin with the numerical coefficients, negative six and three. 06:28.160 --> 06:38.565 The GCF of these two numerical expressions is three. We can factor out either a positive or a negative numerical coefficient 06:38.599 --> 06:50.277 in any situation. The GCF of the variable factors, x squared and x, is x to the first power. 06:50.320 --> 07:04.124 So the GCF of the terms gives us three x. So let's factor out three x as our GCF. 07:04.160 --> 07:08.895 Using the reverse process of the distributive property means that we will divide each term 07:08.960 --> 07:17.804 by three x. So negative six x squared divided by three x leaves us with negative two x. 07:17.840 --> 07:25.712 And three x divided by three x leaves us with one. So we'll have a plus one. And again we 07:25.760 --> 07:31.385 can see that if we were to distribute three x through the grouping symbols, 07:31.440 --> 07:42.195 we would have the same polynomial that we began with, negative six x squared plus three x. 07:42.240 --> 07:55.142 Our next example is x to the fourth minus eight x to the third power. Let's begin by finding the GCF of these two terms. 07:55.200 --> 08:02.916 We begin by looking at the numerical coefficients one and eight. The greatest common factor of one 08:02.960 --> 08:09.856 and eight is just one. The greatest common factor of the variable factors x to the fourth and x to 08:09.920 --> 08:19.232 the third is x to the third. So the GCF is one x to the third or just x to the third. So we'll 08:19.280 --> 08:27.641 write this in factored form by starting with x to the third open parentheses, and now we divide each 08:27.680 --> 08:41.221 of these two terms by x to the third, leaving us with x to the first power, or just x, minus eight. This is the factored form 08:41.254 --> 08:54.501 of this polynomial. Now let's take a look at this polynomial. We have three terms here. So as we look for the GCF 08:54.560 --> 09:01.641 of these three terms, we again begin with the numerical coefficients. 09:01.680 --> 09:09.316 Let's make a list in this case of the factors of each of these numerical coefficients. 09:09.360 --> 09:24.998 The factors of twelve include one, two, three, four, six, and twelve. The factors of twenty 09:25.040 --> 09:39.646 include one, two, four, five, ten, and twenty. And the factors of ten 09:39.680 --> 09:51.040 include one, two, five, and ten. Let's find the factors that we have in common 09:51.057 --> 09:57.797 for these three numerical coefficients. Well, of course they all have a factor of one. 09:57.840 --> 10:06.840 I notice that they also have a factor of two in common. There are no other factors in this list 10:06.880 --> 10:17.551 that are common between all three numerical coefficients. So our numerical GCF is two. 10:17.600 --> 10:24.124 And now looking for the variable factors GCF, we have x squared, x to the first, 10:24.160 --> 10:34.100 and x to the third. So the GCF of this variable factor is x to the first power. We have a y to 10:34.160 --> 10:44.577 the third, y squared, and y to the fourth. So the GCF of this variable factor is y squared. 10:44.640 --> 10:53.887 So the GCF of the three terms of this polynomial is two x y squared. Now let's write this 10:53.920 --> 11:05.565 polynomial in factored form. We begin with the GCF two x y squared followed by an open parenthesis. 11:05.600 --> 11:17.711 And now we divide each of these terms by our GCF two x y squared. Twelve divided by two is six. 11:17.760 --> 11:27.220 x squared divided by x is x. And y to the third divided by y squared is y. 11:27.280 --> 11:36.029 Moving to our next term, negative twenty divided by two is negative ten. x divided by 11:36.080 --> 11:44.738 x is one. y squared divided by y squared is one. So we're just left with minus ten 11:44.800 --> 11:53.646 when we factor out the two x y squared from minus twenty x y squared. And now for the last term, 11:53.680 --> 12:04.657 we divide ten x cubed y to the fourth by two x y squared. Ten divided by two is five. 12:04.720 --> 12:17.070 x cubed divided by x is x squared. And y to the fourth divided by y squared is y squared. 12:17.120 --> 12:27.480 This is factored form of this polynomial. And in our next example, we have y 12:27.520 --> 12:35.555 times x minus four plus three times x minus four. Let's first identify the terms 12:35.600 --> 12:48.435 in this polynomial. We have the term y times x minus four and the term three times x minus four. 12:48.480 --> 12:58.111 I notice here that what they have in common is the factor x minus four. 12:58.160 --> 13:11.891 Each of these terms has a factor of x minus four in common, and that is the GCF of this polynomial. So let's factor 13:11.925 --> 13:26.906 out that GCF, x minus four. We write the GCF followed by an open parenthesis, and now we'll divide each term by that GCF, 13:26.960 --> 13:39.385 x minus four. If I divide y times x minus four by x minus four, that leaves me with y. 13:39.440 --> 13:47.861 And in the second term, if I divide three times x minus four by x minus four, that leaves me with 13:47.920 --> 14:04.911 three or plus three. So factored form of this polynomial is x minus four times y plus three. 14:04.960 --> 14:11.384 Next, we're going to talk about a type of factoring called factoring by grouping. 14:11.440 --> 14:17.457 Let's look at these two examples where we're asked to factor each polynomial. 14:17.520 --> 14:24.497 Sometimes we can factor a polynomial by grouping the terms of the given polynomial 14:24.560 --> 14:32.038 and looking for common factors in those groupings. This is known as factored by grouping. 14:32.080 --> 14:41.014 We often look for polynomials with four terms as possible candidates for this method. Let's consider this polynomial 14:41.047 --> 14:53.559 first with four terms, a b plus six a plus two b plus twelve. We'll begin by grouping this polynomial into groups 14:53.600 --> 15:06.873 of two. So I will group a b plus six a, as one group and two b plus twelve as the second group. Now let's look for 15:06.906 --> 15:21.220 the GCF of our first group, a b and six a. The GCF of the numerical coefficients is one. The GCF of the variable factors 15:21.254 --> 15:36.335 is a. So the GCF of this grouping is a. If I factor an a out of this first group, that will leave me with b plus six. 15:36.400 --> 15:48.347 Now I have a plus sign which we will include here, and we'll look for now the GCF of our second group, two b plus twelve. 15:48.400 --> 15:56.355 The GCF of the numerical coefficients is two, and there is no common factor of the variable factors. 15:56.400 --> 16:05.565 So the GCF of these two terms is two. We will now factor a two out of this grouping. 16:05.600 --> 16:16.509 And when I factor a two out of two b plus twelve, that leaves us with b plus six. Now this 16:16.560 --> 16:24.584 intermediate result looks familiar, similar to the example that we did in the last problem. 16:24.640 --> 16:36.195 Notice now we have this written as two terms, a times b plus six and two times b plus six. 16:36.240 --> 16:44.103 Each of these two terms has the common factor b plus six. So we now can factor the common factor b 16:44.160 --> 16:56.115 plus six out of this expression. So I will factor a b plus six out, and then open parentheses, and 16:56.160 --> 17:05.057 now we divide each term by that common factor to find out what remains. a times b plus six 17:05.091 --> 17:15.301 divided by b plus six is a, and two times b plus six divided by b plus six is plus two. 17:15.360 --> 17:28.814 This is the factored form of this original polynomial. It's important when we're doing grouping to look for ways 17:28.848 --> 17:36.756 that we can get a common factor from the two groupings. Sometimes it's necessary to rearrange terms 17:36.800 --> 17:44.764 or to factor out a different sign positive or negative for our GCF. 17:44.800 --> 17:54.507 Let's take a look at this next example, x cubed minus x squared minus five x plus five. 17:54.560 --> 18:01.881 We'll begin by grouping this into groups of two. But notice, that I have a minus sign 18:01.920 --> 18:11.624 in the front of my third term. So let's rewrite this first as x cubed minus x squared plus 18:11.680 --> 18:22.435 a negative x plus five. And now I'm going to group this polynomial into groups of two. The first 18:22.480 --> 18:32.912 grouping will be x cubed minus x squared, and my second grouping will be negative five x plus five. 18:32.960 --> 18:43.990 The plus sign still remains between these two groupings. Now we look for the GCF of x cubed and x squared, 18:44.023 --> 19:00.506 which is x squared. And if I factor an x squared out of x cubed minus x squared, that leaves me with x minus one. 19:00.560 --> 19:07.079 Now looking at this next grouping, I have negative five x plus five. The numerical coefficient that 19:07.120 --> 19:14.353 we have in common here is five, and there is no variable factor that we have in common. So 19:14.400 --> 19:22.028 five would be a potential candidate for our GCF. But notice what would happen if I factored out a 19:22.080 --> 19:33.639 positive five. This would leave me with a negative x plus one, and what I'm looking for is a positive 19:33.680 --> 19:43.215 x and a negative one. So instead of factoring out a positive five, let's factor out instead 19:43.280 --> 19:56.328 a negative five. Dividing each term here now by negative five leaves me with x minus one. And I see then that 19:56.362 --> 20:09.475 I have two terms with the common factor of x minus one. So we can again factor out the GCF of x minus one. 20:09.520 --> 20:16.015 x minus one is our common factor. And when I divide each term by x minus one, 20:16.080 --> 20:33.840 that will leave me with x squared minus five. This is factored form of the polynomial.