WEBVTT 00:21.600 --> 00:26.660 In this video, we're going to learn about factoring trinomials of the form 00:26.720 --> 00:38.205 x squared plus b x plus c. Let's look at this expression, x minus two times x plus five. If 00:38.240 --> 00:43.477 we multiply these two binomials together, which we have already learned how to do, 00:43.520 --> 00:50.751 we will get the trinomial x squared plus three x minus ten. This form where we have 00:50.800 --> 00:58.826 the two binomials multiplied together is known as factored form. The factored form of a quadratic 00:58.880 --> 01:07.401 expression, as this is, is the product of two linear factors and possibly a constant. 01:07.440 --> 01:15.142 Let's see if we can determine some relationships between this factored form and this trinomial. 01:15.200 --> 01:20.514 As I look at this factored form, I notice that the first terms of my 01:20.560 --> 01:28.255 two binomials multiply together to give me the first term in my trinomial, x squared. 01:28.320 --> 01:36.096 I also notice that these two pieces of my binomials minus two and five are 01:36.160 --> 01:43.804 factors of this last term in my trinomial. They are factors of negative ten. 01:43.840 --> 01:52.746 Further, if I add these two numbers, negative two and five, that is the coefficient that is in my 01:52.800 --> 01:59.620 middle term or my b x term. We can use these patterns when helping us determine whether or 01:59.680 --> 02:13.801 not the trinomial can be factored. We'll look for two numbers whose product is c and whose sum is b. 02:13.840 --> 02:19.807 If a quadratic cannot be factored over the integers, we say that it is prime. Let's 02:19.840 --> 02:28.349 look at some examples. We want to factor each trinomial or state that the expression is prime. 02:28.400 --> 02:35.456 Our first example is x squared plus five x plus six. We notice that this is in the form 02:35.520 --> 02:44.431 x squared plus b x plus c, and there are no common factors in these terms. We always want to look 02:44.480 --> 02:53.140 for a greatest common factor when factoring first, prior to moving into this next step of factoring. 02:53.200 --> 03:02.182 Now to determine how this factors, I'm going to set up two linear factors, and I notice that the 03:02.240 --> 03:08.656 first term in this trinomial is x squared. So my first term of each of these binomials 03:08.720 --> 03:18.265 will be x. When I multiply x times x, we get x squared. To determine the second term of each of 03:18.320 --> 03:26.607 these binomials, I'm going to first look at this constant, six, and I'm looking for two numbers who 03:26.640 --> 03:39.219 are factors of six, and those two numbers must sum to give me five, which is the b value here. 03:39.280 --> 03:49.630 So let's think of all of the different ways we can factor six. I know that one times six gives me 03:49.680 --> 03:59.707 six. But if I add one and six together, I do not get a sum of five. I know that two times three 03:59.760 --> 04:08.816 also gives me six, and when I add two and three, then I do get the correct sum of positive five. So 04:08.880 --> 04:18.992 we will use these two numbers, a positive two and a positive three, to finish our binomial factors. 04:19.040 --> 04:26.033 Now the order of these two linear factors does not matter because multiplication is commutative. 04:26.080 --> 04:32.606 So the factored form of x squared plus five x plus six is x plus two 04:32.640 --> 04:39.079 times x plus three. You can always check that you have factored correctly by using the 04:39.120 --> 04:46.687 distributive property to multiply this out, and you will get the trinomial that you began with. 04:46.720 --> 04:56.196 Let's try another one. In this problem, we have m squared minus seventeen m plus seventy. So I'm 04:56.240 --> 05:03.537 going to set up my two linear factors. Again, I notice that there is no greatest common factor 05:03.600 --> 05:10.511 or no common factor of these three terms. My first factor is, my first term rather is, 05:10.560 --> 05:21.788 m squared. And so the first terms of my binomial will be m and m. m times m gives me m squared. 05:21.840 --> 05:29.396 Now to find the second terms of these binomials, I'm looking for two numbers or two factors 05:29.440 --> 05:37.704 that give me the product of seventy, and they must add together to give me the sum 05:37.760 --> 05:45.312 of negative seventeen. Think about what those factors might be. You might try 05:45.360 --> 05:50.183 one and seventy and realize that that is way too big of a sum. That's seventy-one. 05:50.240 --> 05:56.423 Or two and thirty-five, again you're much too high when you add those numbers together. 05:56.480 --> 06:06.199 Continuing to think through the factors of seventy, you might get to seven and ten. What happens when you add 06:06.233 --> 06:16.343 seven and ten? Well, we get seventeen, a positive seventeen, but what I'm looking for is a negative seventeen. 06:16.400 --> 06:24.017 If I change the sign here of each of those factors, negative seven and negative ten, 06:24.080 --> 06:32.259 I see that the product is still positive seventy and the sum is negative seventeen, 06:32.320 --> 06:39.566 just as we want it. So the second term of each of our binomials will be these two 06:39.600 --> 06:50.744 values, minus seven and minus ten. So the factored form of m squared minus seventeen m plus seventy 06:50.800 --> 07:03.156 is m minus seven times m minus ten. Let's continue our work with factoring. Next, we want to factor a squared 07:03.190 --> 07:12.099 minus two a minus forty-eight. We recognize that this is in the form x squared plus b x plus c, and there are 07:12.160 --> 07:21.942 no common factors in these three terms. So we'll set up our two linear factors. 07:22.000 --> 07:26.279 And as we consider the first term of these linear factors, we look to the first term of our 07:26.320 --> 07:35.222 trinomial which is a squared. That means that the first term of each of our binomials will be a. a 07:35.280 --> 07:43.163 times a gives me a squared. As we consider what might be the second terms of our binomials, we 07:43.200 --> 07:50.504 need to look at the value of c which is negative forty-eight. So I need two numbers that multiply 07:50.560 --> 08:00.046 together to give me negative forty-eight, and they must add together to give me negative two. Now 08:00.080 --> 08:07.154 we're looking for a product that is negative. This means that one of the factors must be positive 08:07.200 --> 08:13.059 while the other factor is negative because a positive times a negative gives us a negative. 08:13.120 --> 08:21.468 So we may start to think of all the ways we can make forty-eight. One and forty-eight, 08:21.520 --> 08:27.874 with one of them being negative, we would choose the larger number to be negative so that the sum 08:27.920 --> 08:35.949 is also negative. But notice that this sum is much too high. This gives us negative forty-seven. Two 08:36.000 --> 08:43.456 and twenty-four, with the twenty-four being negative, still the sum would be much too high. 08:43.520 --> 09:01.374 Three and sixteen, four and twelve, six and eight. And if the eight were negative, then this sum gives us negative two. 09:01.440 --> 09:06.913 So these are the two numbers that finish the binomials in this factored form. 09:06.960 --> 09:15.555 We have a positive six and a negative eight. Again, the order of the two binomials does not matter, so 09:15.600 --> 09:24.364 I could have written this as a minus eight times a plus six. And if I multiply these out using the 09:24.400 --> 09:35.308 distributive property, I will see that I get this trinomial a squared minus two a minus forty-eight. 09:35.360 --> 09:44.985 In our next example, we have x squared plus two x minus ten. As we begin to try to factor this, 09:45.040 --> 09:50.757 we would think that we would have two linear factors and that the first term 09:50.800 --> 09:58.898 of those linear factors would be x in order that I get x squared for my first term. 09:58.960 --> 10:05.705 And then I look to my constant term negative ten. And I'm trying to find two numbers that 10:05.760 --> 10:15.715 multiply together to give me negative ten, and they must add together to give me a positive two. 10:15.760 --> 10:22.355 Let's think of the ways that I can make negative ten by multiplying two numbers together. I can do 10:22.400 --> 10:29.229 one and ten. In this case, if I'm trying to get a positive sum and I want a negative product, 10:29.280 --> 10:35.135 then the larger number would be positive. So I would have a negative one and a positive ten, 10:35.200 --> 10:42.108 and that sum would be positive nine. The only other way that we can make ten 10:42.160 --> 10:50.583 using the product of two numbers is two and five. And again, to get a positive sum, that two 10:50.640 --> 10:57.791 would be negative. But this sum is not positive two. This sum is positive three. 10:57.840 --> 11:04.831 There are no other factors of negative ten. There are no factors that multiply together 11:04.880 --> 11:11.871 to give me negative ten and add together to give me positive two. Thus, this trinomial 11:11.920 --> 11:25.885 can't be factored over the integers. When that is the case, we say that the trinomial is prime. 11:25.920 --> 11:39.466 Let's take a look at another example. Here we have x squared plus x y minus twelve y squared. This trinomial is in the form 11:39.520 --> 11:46.339 x squared plus b x plus c, in this case though my c is a little different looking. 11:46.400 --> 11:55.949 It's negative twelve y squared is my c, and my b value which is the factor in front of the x 11:56.000 --> 12:06.359 is y. We'll begin by setting up the factors of our two binomials the same way we have for the others. 12:06.400 --> 12:13.967 I notice that my first term is x squared in the trinomial, so my first two terms of my binomials 12:14.000 --> 12:23.040 will be x because x times x is x squared. Now in finding the second terms of these binomials, 12:23.042 --> 12:30.116 I need two numbers that multiply together to give me negative twelve y squared, 12:30.160 --> 12:46.599 which is our c value, and they need to add together to give me y or one y. What would I multiply these two numbers together? 12:46.640 --> 12:53.006 If I'm going to get a sum of y, I would like for each of these factors to include y, 12:53.040 --> 12:57.710 so that I can add them together as like terms. So let's really focus on this 12:57.760 --> 13:05.385 negative twelve. What ways can I make negative twelve by multiplying two numbers together? 13:05.440 --> 13:13.726 One and twelve, six and two, three and four. I notice that I have to have different signs to get a 13:13.760 --> 13:20.767 negative product, and I need a positive sum. So the larger of the two numbers will be positive. 13:20.800 --> 13:31.244 So we can use four y and negative three y. The sum of four y and negative three y 13:31.280 --> 13:46.826 is a positive one y. So this will give me x plus four y times x minus three y. 13:46.880 --> 13:56.035 And in our last example here, we have sixteen b plus fifteen plus b squared. 13:56.080 --> 14:03.242 Is this in the form x squared plus b x plus c? Well, it's got the right terms, 14:03.280 --> 14:10.516 but they're not in the right order. So let's begin by rearranging this trinomial. 14:10.560 --> 14:21.828 This can be written as b squared plus sixteen b plus fifteen. Remember that addition is commutative, so I 14:21.861 --> 14:31.804 can rearrange the order of these terms. And now I can recognize that this is in the form x squared plus b x plus c, 14:31.840 --> 14:39.879 where b is sixteen and c is fifteen. So we're going to need two numbers 14:39.920 --> 14:48.020 that multiply together to give me fifteen and their sum is sixteen. 14:48.080 --> 14:59.465 This would be fifteen and one. As I write the factored form with its linear factors, I begin by looking at 14:59.520 --> 15:07.707 the b squared and recognizing that each of these binomials must contain a b to the first power, 15:07.760 --> 15:15.481 and then I use my factors fifteen and one to complete the binomials, leaving me with 15:15.520 --> 15:28.027 b plus fifteen times b plus one as the factored form of this trinomial. 15:28.080 --> 15:35.301 Recall from previous videos and sections that we begin our factoring by looking for a GCF or 15:35.360 --> 15:41.541 greatest common factor as we factor our expressions. So when we're looking 15:41.600 --> 15:46.979 at these examples, the directions are to factor each expression completely. 15:47.040 --> 15:56.422 And the first example is five t squared minus sixty t plus one hundred sixty. Looking at those three terms, you should 15:56.455 --> 16:09.735 notice a common factor of five. The GCF of this expression is five. So we'll begin by factoring out a five. And to 16:09.769 --> 16:15.875 determine what's left in these grouping symbols, we simply divide each of these three terms by five. 16:15.920 --> 16:28.187 This will leave us with t squared minus twelve t plus thirty-two. So we have factored out the GCF, 16:28.240 --> 16:35.628 but we have not yet factored completely. Notice that what is left here is a trinomial of the form 16:35.680 --> 16:44.837 x squared plus b x plus c, and it is possible that this trinomial can be factored. So let's see. 16:44.880 --> 16:52.678 If it can be factored, it's a quadratic, so that means we would have two linear factors. 16:52.720 --> 17:02.188 And the first term of these binomials would be t because t times t gives me t squared. To find 17:02.240 --> 17:08.828 the second term of each of these binomials, I look to my constant term thirty-two, and 17:08.880 --> 17:20.906 I'm looking for factors of thirty-two that sum together to give me negative twelve. 17:20.960 --> 17:26.579 So if I'm looking for two numbers that multiply together to give me a positive number and they 17:26.640 --> 17:35.221 add together to give me a negative number, then those two numbers both need to be negative. Let's 17:35.280 --> 17:41.327 think of the different ways that we can make thirty-two by multiplying two numbers together. 17:41.360 --> 17:49.235 We could do one times thirty-two, or negative one times negative thirty-two, and that sum would be much too large. 17:49.280 --> 17:58.744 We could do two and sixteen. Again, the sum is too large. We could do four and eight. 17:58.800 --> 18:05.151 Four times eight does give me thirty-two, but the sum is positive twelve. And as noted before, 18:05.200 --> 18:10.656 in order to get a negative sum here, I'm going to need to have negative four and negative 18:10.720 --> 18:19.298 eight. That gives me a positive thirty-two as a product and a negative sum of twelve. 18:19.360 --> 18:26.439 So those will be the second terms of each of these binomials. We'll have t minus four times 18:26.480 --> 18:34.046 t minus eight. But let's not forget that we factored out a GCF in the first step. 18:34.080 --> 18:44.523 And we can't leave that out. So we must put that back. And our final factored form is five times t 18:44.560 --> 18:50.196 minus four times t minus eight. We could check this by using the distributive property to 18:50.240 --> 18:59.872 multiply, and we would get this original trinomial. Let's look at our next example. We have three y 18:59.920 --> 19:07.246 cubed minus eighteen y squared minus twenty-one y. Let's look for the 19:07.280 --> 19:14.520 GCF of these three terms. Well, I notice that three, eighteen, and twenty-one have 19:14.560 --> 19:24.263 a common factor of three, and y cubed, y squared, and y have a y in common. So the common factor, 19:24.320 --> 19:36.075 or the greatest common factor, of all three terms is three y. And what's remaining when I factor a three y from 19:36.108 --> 19:51.857 these three terms is y squared minus six y minus seven. So this is a factored form, but it is not completely factored 19:51.891 --> 20:01.233 because I see that I have a trinomial here in these grouping symbols that is a quadratic that may be factored further. 20:01.280 --> 20:07.640 And so if it can be factored further, it will factor into two linear factors. The first 20:07.680 --> 20:16.282 term of each will be y because y times y is y squared. And to come up with the second terms, 20:16.320 --> 20:23.856 we need two numbers that multiply together to give me negative seven, and they must add together 20:23.920 --> 20:31.697 to give me negative six. How do we make a product of a negative seven with two numbers? 20:31.760 --> 20:38.337 One of those numbers has to be negative, and one must be positive. So I know that one and 20:38.400 --> 20:48.580 seven are factors. Which one needs to be negative in order to get a negative sum? 20:48.640 --> 20:55.454 One plus negative seven gives us negative six. So these are our factors. We will have y 20:55.520 --> 21:06.899 plus one and y minus seven as our linear factors. And we must not forget the GCF of three y. So our 21:06.960 --> 21:18.577 completely factored form here of this expression is three y times y plus one times y minus seven. 21:18.640 --> 21:25.784 And here's our last expression, our last example. Negative x squared plus nine x 21:25.840 --> 21:34.993 minus fourteen. The only common factor in this trinomial is one or negative one. 21:35.040 --> 21:40.766 In general, when your trinomial or expression begins with a negative number, 21:40.800 --> 21:48.340 it's easiest to factor if you first factor out a negative one. So we're going to begin there. 21:48.400 --> 21:56.415 We will factor out a negative one, and then this will give us x squared minus nine x 21:56.480 --> 22:03.455 plus fourteen as the remaining trinomial inside those grouping symbols. 22:03.520 --> 22:13.565 It's a quadratic, and so we can factor that quadratic with two linear factors. The first term 22:13.600 --> 22:20.339 of each will be x. And for the last term I need two numbers that multiply together to give me 22:20.400 --> 22:28.180 positive fourteen, and they must add together to give me negative nine. What two numbers multiply 22:28.240 --> 22:37.322 together to give me positive fourteen and add together to give me negative nine? 22:37.360 --> 23:00.012 How about negative two and negative seven? So our completely factored form will be negative one times x minus two times x minus seven.