Algebra 2
Hello, I'm Kirk weiler, and this is common core algebra two. By E math instruction. Today, we're going to be doing unit 6 lesson number three on factoring trinomials. Now in the last lesson, we reviewed two types of factoring. That was how to factor out a greatest common factor. And also how to factor the difference of perfect squares. Today, we're going to tackle arguably the hardest type of factoring, which is factoring trinomials. All right? So let's jump right into that. Now, before we factor trinomials, what we're going to be doing is we're going to write each we're going to get a little practice on, I don't want to call it the opposite of factoring, but sort of the opposite process, which is multiply. Multiplying binomials. So it says without using your calculator, write each of the following products in simplest AX squared plus BX plus C form. Now, technically speaking, the way that one of these products is done is by what I would call double distribution. In other words, I'm going to multiply three X by 5 X plus 7. And I'm going to add two times 5 X plus 7. So in other words, both of these quantities distribute into here or better yet, this distributes to both of those quantities. Then distributing again, giving me 15 X squared plus 21 X plus ten X plus 14. Now notice we've got four terms here. And before we move on, let's talk about those a little bit. Where did the 15 X squared come from? That came specifically, let me go in red. The 15 X squared, specifically came from multiplying three X by 5 X. Now that's very important. On the other hand, the 14 specifically came from multiplying the two and the 7. These other two terms, which eventually just get combined into a 31 X, they come from the fact that I've multiplied two by 5 X to give me ten X and three X by 7 to get me 21 X, right? Now our final answer of 15 X squared plus 31 X plus 14 isn't all that remarkable isn't all that important. The important thing to understand and let's talk about this as we go on to letter B is that really we have these first two terms which multiply together, gives me four X squared. These last two terms that multiply together, they give me negative 15. But then we have an inner set of terms, which gives me negative 6 X and an outer set of terms, which gives me positive ten X, which then will always combine. Nothing combines with that four X squared, nothing combines with that negative 15. But these two always combine and give us that sort of center term. All right. Many times students use what's known as a foil process to multiply binomials. First, outer inner last. All right. Any way that you do it, why don't you pause the video now and take care of C and D? All right, let's go through them. So in letter C I use the right side of the pen. We're going to get 5 X times X 5 X squared. I'll get negative four times X negative four X 5 X times negative two, negative ten X and then negative four times negative two would be a positive 8. Again, nothing combines with these two terms. But these two terms in this case combine to be a negative 14 X plus 8. Remember at that point you're just combining like terms. Finally, let's do this one. Four X times three X is 12 X squared. Three times three X is a positive 9 X four times negative 8 is a negative 32 X and three times negative 8 is negative 24. Again, these two combine. To give us a net negative 23 X and then -24. Now, when we factor, what we're going to be looking to do is start with all the things that are circled and go backwards. To these products. And that is much, much harder. But we're going to learn methods to do it in this lesson and then even more in the next lesson. So pause the video now and write down anything you have to before we start to factor. Okay. Let's clear it. Let's start to factor. Okay. So I want to take this trinomial and break it up as the product of two binomials. And I've given four guesses here, because basically, let's just cut right to the chase. In my opinion, there's only one real way to factor trinomials. Guess and check. If your teacher shows you another way that works with products and sums and magical boxes and things that just disappear, hey, fair enough, right? That's up to a teacher's discretion. But to me, the best way to factor a trinomial is to guess and check. So let's start working on that. All right? Below are four guesses of how this trinomial factors. Two of these guesses are unintelligent, meaning they should not even be checked. Cross them out and explain below below them why they are unreasonable. Think about this for a second. All right, let's talk about them. Well, I claim that this is an unreasonable guess. And the reason it's unreasonable is that X times X is X squared, but I need 6 X squared. So that one's out. I don't have to check anything else. Nothing's going to combine with an X squared to give me a 6 X squared. We kept trying to emphasize that in the last problem. I also claim that this one makes no sense. Now granted three X times two X is 6 X squared. But negative two times negative three, that would give me a positive 6, but I've got to have a negative 6. A negative 6. So that one's out. All right? These two are both reasonable guesses. And let's again talk about that for a second. These are reasonable because three X times two X is 6 X squared. And two times negative three is negative 6. So the first term, the first term matches, and the last term matches. Likewise, 6 X times X is 6 X squared. And one times negative 6 is negative 6. So again, that's a reasonable guess. But one of them is correct and one of them is not correct. All based on this negative 35. And it's this simple. It's this simple to figure out which one is correct. We already know the first term works. And we already know the last term works. So all I need to do is check whether this product plus this product sums to be this. And the answer on this one is a resounding no. Let's check this one. 6 X plus one. Times X -6. Right here. Again, I don't need to check the first and the last. I just have to see if this product one X plus this product negative 36 X gives me the right answer. Negative 35 X yes. So when I talk about guessing and checking a lot of students say, I don't want to guess and check. Then I gotta foil the whole thing out. At to multiply these two binomials all the way out. But you don't. If you make intelligent guesses, IE, you make guesses where the first terms multiplied give you the first term of the trinomial. And the last terms multiplied give you the last term, then all you have to do is sort of check the inner and the outer terms and see if they sum. To be that second term of the trinomial. Lots of words flying around. Pause the video now and write this down. This is very, very important. Okay, let's clear out the text. Let's start to factor some trinomials. So I'll guess and check. All right. And we're going to start easy, and then we're going to get a little bit harder, and a little bit harder. Now this first set of four trinomials. Is definitely the easiest. And also very, very important. But they're the easiest because the leading coefficients are all one. And that makes my life a lot easier because immediately I know this has to be an X and this has to be an X now the issue is, of course, looking at this 35. Now we know that 35 is 7 times 5 and one times 35. So these are both reasonable guesses. 7, 5. And now what I need is I need positives and negatives. Well, let me make an incorrect guess first. This is an incorrect guess. But it's a reasonable guess because X times X is X squared and negative 7 times positive 5 is negative 35. It's not the correct guess because when I multiply these two together and I multiply these two together and add them, I get negative two X, but I'm supposed to get a positive two X so that's the wrong guess. Right? It's just wrong. And we can put a giant red line through it. Even extended right there. That's weird. Anyway, so then I'll try this guess. X times 7, X -5. Here I'll get a positive 7 X, here I'll get a negative 5 X, and when I combine them, I get a positive two X and that's what I needed. Let me circle this one in green. Right? But that's it. That's the whole deal. Nothing more than that. And again, a lot of students don't like this because it requires throwing down a guess. Checking it, and if it doesn't work, doing it again. But to not like that and define little gimmicky ways around it with products and sums and things like that. Well, you know, I apologize if this seems a little bit harsh, but well then you don't like problem solving. Because that's what it is. It's problem solving is all about taking chances, seeing if they're right and if they're not, going back and taking another shot at it. That's just the way it is. So for instance, letter B, right? I see X squared plus 11 X plus 24. And my first sort of instinct is a hot 24, four and 6. Right? But the problem is plus four X plus 6 X gives me ten X no. I need 11 X ah, maybe X plus three times X plus 8. Let's see, three X 8 X add them together. We get 11 X, that's what I needed. Again, let me circle ingredients so we know what the right answer is. That's it. All right, why don't you pause the video and try C and D? All right, let's do it. Now you can certainly use and develop some number sense on here, which is helpful. So for instance, right? Based on the fact that this is a positive 22, the way this factors either has to look like this, or it has to look like this because these two things multiplied after give me a positive. So they either broke that to be negative or both have to be positive. But this is negative. Right? Therefore, if we're going to start off with a guess that has a reasonable shot of being correct, we start with that. And then we see negative 11 X, negative two X, those combined to be negative 13 X, that checks. And that means this. Is the correct factorization. All right, this guy. Again, you can also develop some good sense of things, right, 5, and ten, two, and 25, et cetera. Probably 5 and ten. We know one's going to be negative and one's got to be positive because this is negative. And we can also develop a sense that the larger one has to be the negative because they have to combine to be a net negative. And this one does end up being correct. I'm not going to go through the check. Hopefully you got that one right. Now again, these were easy because I didn't have to think very much about the coefficients on the X terms because they had to both be one. The next level is going to get a little bit harder. So don't fall into some really easy patterns and go, oh yeah, man, I got this factoring trinomial business down. Pause the video and think about what we've done, and then let's clear it out. Okay. Exercise three. Using a guess and check technique, factor each of the following trinomials that have prime leading coefficients show each check, guess and check. Now, I think that these as being sort of the second easiest or the second hardest, depending on how you want to look at it. And what I mean by that is ones where the leading coefficient on X squared are both prime numbers. Okay? Now, the reason that I think that these are still fairly easy. Is that again, I have no choice. It's got to be three X and X it just has to be. But having a coefficient there other than one now starts to throw things off a little bit. Okay? Now let's talk about 40. It's got to be negative. But what do we have? We got two and 20. We've got tough. Four and ten, we've got 5 and 8, all right? And I kind of do want to walk down the road of at least one wrong guess. So let me put a ten here and let me put a four here, and let's go negative. And paused. If one's got to be negative one's got to be positive. So this is a reasonable guess again, because three X times X is three X squared. And ten times negative four is negative 40. So it's reasonable. It's not right. Because when I multiply these two together, I get ten X when I multiply these two together, I get negative 12 X and combined, those will be negative two X so, nope, that's not right. So let's try another one. Now, it could be the ten and the four, but maybe they're just switched, maybe the four goes with the three, and the ten goes with the X all right, but I'm going to go with a 5 and an 8. I'm going to go 5 here, like O 8 here, one's got to be positive one's got to be negative. And again, it's reasonable three X times X is three X squared, negative 5 times positive 8 is negative 40. Let's see if it's right. Here I'll get negative 5 X here I'll get positive 24 X and when I add those together, I get a net positive 19 X and that's exactly what I needed. All right. The key is to try it as many times as you need to until you get it right. How about you pause the video now and you try this one. All right, now in certain ways, this problem is a bit easier and let me explain why. All right, it's not about two versus three. That's pretty irrelevant. But it's about the fact that this is a positive product, but a negative sum, and that means both of these things have to be negative. Now here's where, of course, it's tough, because 18 is two times 9. One times 18, three times 6, and that's it. But what goes with what? Maybe it's this. Right again, reasonable two X times X is two X squared. Negative two times negative 9 is positive 18. Let's see. I'm going to get negative two X there. Negative 18 X there. And those two combined with addition to be negative 20 X, I need negative 15 X so no. Eh. But I do know that they're both negative. So let's try three and 6. Get rid of that. Get rid of that. Get rid of that. Let's see, here I'd have negative three X and I remember it's two times negative 6. That's going to be negative 12 X when I add negative three X negative 12 X I got negative 15 X and that's what I needed. I mean, again, I could make hundreds of incorrect guesses and just keep checking, checking, checking, checking. But we won't do too much of that simply because we're trying to get you the idea, not bang our heads in the wall guessing and then failing. All right? So pause the video, write down anything you need to, and then we'll factor some more. Okay. Let's clear it out. And move it on. All right, now the absolute hardest problems. Absolute hardest trinomials to factor. R ones where the leading coefficient are not prime. Not prime. So that's tough. That can be very difficult. Because first off, that 15 could be one in 15. It could be three and 5, right? This is much more difficult. Now, what should you start with? I don't know. Why don't we start with X and 15 X? Let's see how that works. Luckily for us, the two is not too bad, right? I don't know. Maybe put the two here and the one here. Now they both have to be positive because we have a positive product in a positive sum. So they can't be negative. So maybe that's it. What will we get here? Let's see, 15 X, two X, remember we're adding these. We get 17 X, we need 13 X all right, so that's not right. We could flip flop these, but we wouldn't find anything there either. So let's try three X plus two. And 5 X, I guess it would have to be plus one. Again, it's a reasonable guess three X times 5 X is 15 X squared, two times one is two. So here we'll have ten X here we'll have 13 X, I'm going to add them together. I get 13 X checks. All right, pause the video and take a shot at letter B. One thing I like about factoring problems is you should know whether you have them right or wrong. So ten, ten, go back to blue. Ten is either one times ten. Or 5 times two. Um. Well, let's go with ten X and X let me give me ten X squared. Oh, 30. 30 is tough because 30 is a lot of different things. It's one times 30. It's two times 15. It's three times ten. It's 5 times 6. And that's it. Oh, man. I don't know. Let's just take a shot. Let's go with this. One's got to be negative. One's got to be positive. Why not? So let's see, we have 5 X -60 X is going to be negative 55 X no. I need a positive 13 X so that doesn't even come close. Let's try 5 X. And two X let's go with a 6 and a 5. Let's see if this works. Negative 12 X positive 25 X have those together. You have positive 13 X we got. Again, I apologize for not showing more incorrect guesses, but we'll see a tech meet tomorrow that will allow us to, well, in the next lesson, sorry about that. We'll see, we'll see a technique that will allow us to make even better guesses, eliminate even more possibilities so that we're closer to the correct one. Pause the video now, write this down, and then we'll try two more. Okay, let me clear that out. All right. Well, last two, so I'd like you to pause the video, work on both of those, and then we'll go through them. All right. Let's tackle. Now again, if these were frustrating for you, I get it, especially with the 36, 36, one in 36, two and 18, three and 12, four and 9, 6 and 6. I mean, there's a lot of them for 36. So again, we could go about doing many guesses and many checks. Ultimately speaking, the correct guess and check on this one. Is 6 X two X -5 and positive three. And again, let's talk about reasonableness, 6 X and two X, 12 X squared, negative 5 positive three, negative 15. But again, here's the kicker. They're all going to negative ten X here, I'll get a positive 18 X, and when I add them together, I get a positive 8 X, so that's got to be correct. This one, which probably would be the one that would take the most guesses. As 9 X four X two and three. This is negative 8 X, negative 27 X when you combine the two. Yeah, negative 35 X so it checks. I know these can be frustrating. They can sometimes require a lot of guessing and checking. But if you understand how to make an intelligent guess, then the check really should be quite quick. All right. Well, pause the video now. We're going to clear out the text and finish up. Okay, here we go. We're going to need to be able to factor fluently throughout this course. Throughout the course. And factoring trinomials is a critical piece of that fluency. All it takes is hard work. And perseverance. Every one of us has that inside of us, not just for factoring trinomials, but for every problem that we tackle, both in math and in life. The question is, how long can we keep polling on that perseverance before we give up? Don't give up when you're factoring trinomials. You will get them. Extra sheets of paper. That helps too. All right. So thank you for joining me. For another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. And keep solving problems.
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