Video transcript
In this video recording I want to do a bunch of examples of factoring a moment degree polynomial, which is much called a quadratic equation. sometimes a quadratic polynomial, or merely 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 baron. In this case, in all of the examples we ‘ll do, it ‘ll be ten. So let ‘s say I have the quadratic formulation, adam squared plus 10x, plus 9. And I want to factor it into the merchandise of two binomials. How do we do that ? good, let ‘s just think about what happens if we were to take x plus a, and multiply that by x plus bacillus. If we were to multiply these two things, what happens ? Well, we have a little bite of know doing this. This will be ten times x, which is ten squared, plus x times b, which is bx, plus a times x, plus a time barn — plus ab. Or if we want to add these two in the middle right here, because they ‘re both coefficients of adam. We could right this as ten squared plus — I can write it as bacillus plus a, or a summation barn, ten, plus ab. so in general, if we assume that this is the product of two binomials, we see that this middle coefficient on the ten term, or you could say the inaugural degree coefficient there, that ‘s going to be the sum of our a and boron. 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 lapp thing as this. so can we somehow traffic pattern match this to that ? Is there some a and boron where a plus b is adequate to 10 ? And a times bacillus is equal to 9 ? good, let ‘s just think about it a little bit. What are the factors of 9 ? What are the things that a and b-complex vitamin could be equal to ? And we ‘re assuming that everything is an integer. And normally when we’re factorization, specially 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. immediately, if it ‘s a 3 and a 3, then you ‘ll have 3 plus 3 — that does n’t adequate 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 ten 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. therefore when you see something like this, when the coefficient on the adam squared term, or the head coefficient on this quadratic is a 1, you can good say, all proper, 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 criterion form. Or if it ‘s not in standard form, you should put it in that shape, so that you can constantly say, OK, whatever ‘s on the foremost degree coefficient, my a and bel have to add to that. Whatever ‘s my constant term, my a time b, the product has to be that. Let ‘s do respective 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 unlike number — x squared plus 15x, plus 50. And we want to factor this. Well, same drill. We have an ten squared term. We have a first degree terminus. This right here should be the union of two numbers. And then this terminus, the changeless terminus 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 snatch 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 bel 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. therefore 1 plus 50 does not add up to 15. 2 plus 25 does not add astir 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 extinct, and see that this is indeed ten squared plus 15x, plus 10. In fact, let ‘s do it. adam times x, adam squared. x times 10, plus 10x. 5 times adam, 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 ten squared plus 15x, plus 50. Let ‘s up the stakes a fiddling bit, introduce some negative signs in here. Let ‘s say I had x squared minus 11x, plus 24. now, it ‘s the accurate like rationale. I need to think of two numbers, that when I add them, need to be equal to negative 11. a asset b need to be adequate to negative 11. And a times b need to be equal to 24. nowadays, there ‘s something for you to think about. When I multiply both of these numbers, I ‘m getting a positive numeral. 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 means I ‘m going to get a positive issue hera. 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 intersection is positive, tells me that both a and b are minus. a and b have to be veto. Remember, one ca n’t be negative and the other one ca n’t be positive, because the product would be negative. And they both ca n’t be positive, because when you add them it would get you a positive phone number. So let ‘s barely 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. nowadays, which of these when I multiply these — well, obviously when I multiply 1 times 24, I get 24. When I get 2 times 11 — good-for-nothing, 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 minus 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 does n’t quite exploit out, right ? Because we have a damaging 11 here. But what if we did negative 3 and negative 8 ? negative 3 times damaging 8 is peer to positive 24. minus 3 plus negative 8 is peer to negative 11. indeed negative 3 and minus 8 work. so if we factor this, x squared minus 11x, plus 24 is going to be adequate 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 intersection of my two numbers is negative, right ? a fourth dimension b is equal to negative 14. My product is negative. That tells me that one of them is cocksure, and one of them is damaging. And when I add the two, a plus bacillus, it ‘d be peer to 5. So let ‘s think about the factors of 14. And what combinations of them, when I add them, if one is plus and one is negative, or I ‘m very kind of taking the remainder of the two, do I get 5 ? thus if I take 1 and 14 — I ‘m just going to try out things — 1 and 14, negative 1 plus 14 is veto 13. minus 1 plus 14 is 13. indeed let me write all of the combinations that I could do. And finally your brain will merely zone in on it. So you ‘ve got minus 1 plus 14 is equal to 13. And 1 plus negative 14 is adequate to negative 13. thus those do n’t work. That does n’t peer 5. now what about 2 and 7 ? If I do negative 2 — let me do this in a unlike 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 veto 5, so that would n’t have worked. But negative 2 plus 7 works. And negative 2 times 7 is negative 14. indeed there we have it. We know it ‘s adam minus 2, times x plus 7. That ‘s pretty neat. negative 2 times 7 is damaging 14. negative 2 plus 7 is positive 5. Let ‘s do several more of these, just to truly get well honed this skill. So lashkar-e-taiba ‘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 minus 56. And their difference, because one is going to be convinced, and one is going to be negative, right ? Their difference has to be damaging 1. And the numbers that immediately jump out in my genius — and I do n’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 besides 28 times 2. It ‘s all sorts of things. But 8 times 7 in truth jumped out into my brain, because they ‘re identical close to each other. And we need numbers that are very close to each other. And one of these has to be positivist, 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 adequate to negative 56. And then if we take negative 8 plus 7, that is equal to damaging 1, which is precisely the coefficient mighty there. So when I factor this, this is going to be ten minus 8, times x plus 7. This is frequently 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 veto signs, see which of those factors when one is positive, one is negative, add up to the coefficient on the ten term. But as you do more and more practice, you ‘ll see that it ‘ll become a bit of second nature. immediately let ‘s step up the stakes a little snatch more. Let ‘s say we had veto x squared — everything we ‘ve done so far had a convinced coefficient, a positive 1 coefficient on the x squared term. But permit ‘s say we had a damaging adam 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 merely like the problems we ‘ve been doing earlier. So this is the same thing as negative 1 times positive ten 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, lapp 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 plus 23, if it was the other way around, it ‘d be damaging 23. Does n’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 total would be 10. If the 12 is negative, their sum would be negative 10. hush does n’t work. 3 and 8. If the 3 is damaging, their union will be 5. So it works ! sol if we pick negative 3 and 8, negative 3 and 8 work. Because veto 3 plus 8 is 5. negative 3 times 8 is negative 24. so this is going to be equal to — ca n’t forget that negative 1 out movement, and then we factor the inside. negative 1 times adam minus 3, times x plus 8. And if you truly wanted to, you could multiply the negative 1 times this, you would get 3 minus x if you did. Or you do n’t have to. Let ‘s do one more of these. The more practice, the better, I think. All correct, let ‘s say I had minus x squared plus 18x, minus 72. thus once again, I like to factor out the negative 1. therefore this is equal to negative 1 times ten squared, minus 18x, plus 72. now we barely have to think of two numbers, that when I multiply them I get positive 72. So they have to be the same signboard. And that makes it easier in our head, at least in my promontory. When I multiply them, I get positive 72. When I add them, I get negative 18. So they ‘re the same polarity, and their union 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, possibly you think of 8 times 9, but 8 times 9, or veto 8 subtraction 9, or negative 8 plus negative 9, does n’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 damaging 6 plus negative 12, that is equal to veto 18. Notice, it ‘s a bit of an art. You have to try the different factors here. so this will become negative 1 — do n’t want to forget that — times x minus 6, times x minus 12.
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