Kan ikke få nok av at Sal faktoriserer enkle kvadratuttrykk? Her er en håndfull eksempler, bare for deg! Opprettet av Sal Khan og CK-12 Foundation.
- In this video I want to do a bunch of examples of factoring a second degree polynomial, which is often called a quadratic. Sometimes a quadratic polynomial, or just a quadratic itself, or quadratic expression, but all it means is a second degree polynomial. So something that's going to have a variable raised to the second power. In this case, in all of the examples we'll do, it'll be x. So let's say I have the quadratic expression, x squared plus 10x, plus 9. And I want to factor it into the product of two binomials. How do we do that? Well, let's just think about what happens if we were to take x plus a, and multiply that by x plus b. If we were to multiply these two things, what happens? Well, we have a little bit of experience doing this. This will be x times x, which is x squared, plus x times b, which is bx, plus a times x, plus a times b-- plus ab. Or if we want to add these two in the middle right here, because they're both coefficients of x. We could right this as x squared plus-- I can write it as b plus a, or a plus b, x, plus ab. So in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the x term, or you could say the first degree coefficient there, that's going to be the sum of our a and b. And then the constant term is going to be the product of our a and b. Notice, this would map to this, and this would map to this. And, of course, this is the same thing as this. So can we somehow pattern match this to that? Is there some a and b where a plus b is equal to 10? And a times b is equal to 9? Well, let's just think about it a little bit. What are the factors of 9? What are the things that a and b could be equal to? And we're assuming that everything is an integer. And normally when we're factoring, especially when we're beginning to factor, we're dealing with integer numbers. So what are the factors of 9? They're 1, 3, and 9. So this could be a 3 and a 3, or it could be a 1 and a 9. Now, if it's a 3 and a 3, then you'll have 3 plus 3-- that doesn't equal 10. But if it's a 1 and a 9, 1 times 9 is 9. 1 plus 9 is 10. So it does work. So a could be equal to 1, and b could be equal to 9. So we could factor this as being x plus 1, times x plus 9. And if you multiply these two out, using the skills we developed in the last few videos, you'll see that it is indeed x squared plus 10x, plus 9. So when you see something like this, when the coefficient on the x squared term, or the leading coefficient on this quadratic is a 1, you can just say, all right, what two numbers add up to this coefficient right here? And those same two numbers, when you take their product, have to be equal to 9. And of course, this has to be in standard form. Or if it's not in standard form, you should put it in that form, so that you can always say, OK, whatever's on the first degree coefficient, my a and b have to add to that. Whatever's my constant term, my a times b, the product has to be that. Let's do several more examples. I think the more examples we do the more sense this'll make. Let's say we had x squared plus 10x, plus-- well, I already did 10x, let's do a different number-- x squared plus 15x, plus 50. And we want to factor this. Well, same drill. We have an x squared term. We have a first degree term. This right here should be the sum of two numbers. And then this term, the constant term right here, should be the product of two numbers. So we need to think of two numbers that, when I multiply them I get 50, and when I add them, I get 15. And this is going to be a bit of an art that you're going to develop, but the more practice you do, you're going to see that it'll start to come naturally. So what could a and b be? Let's think about the factors of 50. It could be 1 times 50. 2 times 25. Let's see, 4 doesn't go into 50. It could be 5 times 10. I think that's all of them. Let's try out these numbers, and see if any of these add up to 15. So 1 plus 50 does not add up to 15. 2 plus 25 does not add up to 15. But 5 plus 10 does add up to 15. So this could be 5 plus 10, and this could be 5 times 10. So if we were to factor this, this would be equal to x plus 5, times x plus 10. And multiply it out. I encourage you to multiply this out, and see that this is indeed x squared plus 15x, plus 10. In fact, let's do it. x times x, x squared. x times 10, plus 10x. 5 times x, plus 5x. 5 times 10, plus 50. Notice, the 5 times 10 gave us the 50. The 5x plus the 10x is giving us the 15x in between. So it's x squared plus 15x, plus 50. Let's up the stakes a little bit, introduce some negative signs in here. Let's say I had x squared minus 11x, plus 24. Now, it's the exact same principle. I need to think of two numbers, that when I add them, need to be equal to negative 11. a plus b need to be equal to negative 11. And a times b need to be equal to 24. Now, there's something for you to think about. When I multiply both of these numbers, I'm getting a positive number. I'm getting a 24. That means that both of these need to be positive, or both of these need to be negative. That's the only way I'm going to get a positive number here. Now, if when I add them, I get a negative number, if these were positive, there's no way I can add two positive numbers and get a negative number, so the fact that their sum is negative, and the fact that their product is positive, tells me that both a and b are negative. a and b have to be negative. Remember, one can't be negative and the other one can't be positive, because the product would be negative. And they both can't be positive, because when you add them it would get you a positive number. So let's just think about what a and b can be. So two negative numbers. So let's think about the factors of 24. And we'll kind of have to think of the negative factors. But let me see, it could be 1 times 24, 2 times 11, 3 times 8, or 4 times 6. Now, which of these when I multiply these-- well, obviously when I multiply 1 times 24, I get 24. When I get 2 times 11-- sorry, this is 2 times 12. I get 24. So we know that all these, the products are 24. But which two of these, which two factors, when I add them, should I get 11? And then we could say, let's take the negative of both of those. So when you look at these, 3 and 8 jump out. 3 times 8 is equal to 24. 3 plus 8 is equal to 11. But that doesn't quite work out, right? Because we have a negative 11 here. But what if we did negative 3 and negative 8? Negative 3 times negative 8 is equal to positive 24. Negative 3 plus negative 8 is equal to negative 11. So negative 3 and negative 8 work. So if we factor this, x squared minus 11x, plus 24 is going to be equal to x minus 3, times x minus 8. Let's do another one like that. Actually, let's mix it up a little bit. Let's say I had x squared plus 5x, minus 14. So here we have a different situation. The product of my two numbers is negative, right? a times b is equal to negative 14. My product is negative. That tells me that one of them is positive, and one of them is negative. And when I add the two, a plus b, it'd be equal to 5. So let's think about the factors of 14. And what combinations of them, when I add them, if one is positive and one is negative, or I'm really kind of taking the difference of the two, do I get 5? So if I take 1 and 14-- I'm just going to try out things-- 1 and 14, negative 1 plus 14 is negative 13. Negative 1 plus 14 is 13. So let me write all of the combinations that I could do. And eventually your brain will just zone in on it. So you've got negative 1 plus 14 is equal to 13. And 1 plus negative 14 is equal to negative 13. So those don't work. That doesn't equal 5. Now what about 2 and 7? If I do negative 2-- let me do this in a different color-- if I do negative 2 plus 7, that is equal to 5. We're done! That worked! I mean, we could have tried 2 plus negative 7, but that'd be equal to negative 5, so that wouldn't have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. So there we have it. We know it's x minus 2, times x plus 7. That's pretty neat. Negative 2 times 7 is negative 14. Negative 2 plus 7 is positive 5. Let's do several more of these, just to really get well honed this skill. So let's say we have x squared minus x, minus 56. So the product of the two numbers have to be minus 56, have to be negative 56. And their difference, because one is going to be positive, and one is going to be negative, right? Their difference has to be negative 1. And the numbers that immediately jump out in my brain-- and I don't know if they jump out in your brain, we just learned this in the times tables-- 56 is 8 times 7. I mean, there's other numbers. It's also 28 times 2. It's all sorts of things. But 8 times 7 really jumped out into my brain, because they're very close to each other. And we need numbers that are very close to each other. And one of these has to be positive, and one of these has to be negative. Now, the fact that when their sum is negative, tells me that the larger of these two should probably be negative. So if we take negative 8 times 7, that's equal to negative 56. And then if we take negative 8 plus 7, that is equal to negative 1, which is exactly the coefficient right there. So when I factor this, this is going to be x minus 8, times x plus 7. This is often one of the hardest concepts people learn in algebra, because it is a bit of an art. You have to look at all of the factors here, play with the positive and negative signs, see which of those factors when one is positive, one is negative, add up to the coefficient on the x term. But as you do more and more practice, you'll see that it'll become a bit of second nature. Now let's step up the stakes a little bit more. Let's say we had negative x squared-- everything we've done so far had a positive coefficient, a positive 1 coefficient on the x squared term. But let's say we had a negative x squared minus 5x, plus 24. How do we do this? Well, the easiest way I can think of doing it is factor out a negative 1, and then it becomes just like the problems we've been doing before. So this is the same thing as negative 1 times positive x squared, plus 5x, minus 24. Right? I just factored a negative 1 out. You can multiply negative 1 times all of these, and you'll see it becomes this. Or you could factor the negative 1 out and divide all of these by negative 1. And you get that right there. Now, same game as before. I need two numbers, that when I take their product I get negative 24. So one will be positive, one will be negative. When I take their sum, it's going to be 5. So let's think about 24 is 1 and 24. Let's see, if this is negative 1 and 24, it'd be positive 23, if it was the other way around, it'd be negative 23. Doesn't work. What about 2 and 12? Well, if this is negative-- remember, one of these has to be negative. If the 2 is negative, their sum would be 10. If the 12 is negative, their sum would be negative 10. Still doesn't work. 3 and 8. If the 3 is negative, their sum will be 5. So it works! So if we pick negative 3 and 8, negative 3 and 8 work. Because negative 3 plus 8 is 5. Negative 3 times 8 is negative 24. So this is going to be equal to-- can't forget that negative 1 out front, and then we factor the inside. Negative 1 times x minus 3, times x plus 8. And if you really wanted to, you could multiply the negative 1 times this, you would get 3 minus x if you did. Or you don't have to. Let's do one more of these. The more practice, the better, I think. All right, let's say I had negative x squared plus 18x, minus 72. So once again, I like to factor out the negative 1. So this is equal to negative 1 times x squared, minus 18x, plus 72. Now we just have to think of two numbers, that when I multiply them I get positive 72. So they have to be the same sign. And that makes it easier in our head, at least in my head. When I multiply them, I get positive 72. When I add them, I get negative 18. So they're the same sign, and their sum is a negative number, they both must be negative. And we could go through all of the factors of 72. But the one that springs up, maybe you think of 8 times 9, but 8 times 9, or negative 8 minus 9, or negative 8 plus negative 9, doesn't work. That turns into 17. That was close. Let me show you that. Negative 9 plus negative 8, that is equal to negative 17. Close, but no cigar. So what other ones are there? We have 6 and 12. That actually seems pretty good. If we have negative 6 plus negative 12, that is equal to negative 18. Notice, it's a bit of an art. You have to try the different factors here. So this will become negative 1-- don't want to forget that-- times x minus 6, times x minus 12.