CA Algebra I: Factoring quadratics

All right, we're on problem 44. And they said, which is the factored form of 3a squared minus 24ab plus 48b squared? And if this confuses you with a's and b's instead of an x there, I just like to think of in this case, this looks like an x squared. So I like to think of it the same way I would think of it in terms of if this was an x squared minus some number times x plus some other number. So let's put that in the back of your head. If that's confuses you, don't worry about it. But the first thing I like to do is try to get rid of the coefficient on the, in this case, the a squared term. Let's see if we can factor out a number. Well, sure. All of these are divisible by 3. 3 is divisible by 3, 24 is divisible by 3 and 48 is divisible by 3. So we should be able to factor out a 3, so let's do that. So that is equal to 3 times a squared minus 8ab. 24 divided by 3 is 8 and there's a minus in front of it. Plus 48 divided by-- 3. My brain was getting ahead of me. 48 divided by 3 is 16b squared. Now to factor this we have to think about it as, are there two numbers-- well, think of it. Let me rewrite this a little bit. If we view a as kind of the independent variable-- you wouldn't have to do it, but I just wanted you to visualize it properly. 3a squared minus 8b times a plus 16b squared. So if we viewed a squared as kind of the independent variable or the x term, so now this kind of has the shape of polynomials that hopefully you're used to factoring a little bit. We just have to think, OK, are there two numbers that add up to minus 8b, and that when I multiply the two numbers, I get 16b squared? So first all, the number is going to be in terms of b, obviously. Because if I'm adding them and I get a b and-- I get b's. And if I square them I get a b squared. And it also makes me lead to it's the same number, just used twice. Otherwise, it would be weird to get a squared here. So the number that should pop out, you say, OK, what two numbers add up to minus 8 and that when I square it is equal to 16? Minus 4, right? Minus 4, minus 4 is minus 8. Minus 4 squared is 16. So the number we're talking about is minus 4b, right? Minus 4b. b, that's a b. Plus minus 4b is equal to minus 8b. And minus 4b squared is equal to 16b squared. So we can factor this out to be 3-- let me switch colors. 3 times. We said minus 4b is the number. When you add it twice you get minus 8b. And you square it, you get that. So it's a minus 4b times a minus 4b. And that is choice C. Next problem, 40-- did I skip 45? I did. Let me copy and paste 45 in here. Can't skip problems. 45. Copy and paste it in. OK. Maybe over do on top of this one. Let me see if I can erase this. Look at that. All right, now I can paste 45 and we're ready to go. Which is a factor of x squared minus 11x plus 24? So there's two ways to do it. You could just factor this. Well, that's the easiest way. Or you could test for 0's. But we'll just factor. So if we wanted to say-- well, let me just-- x squared minus 11x plus 24. So just like the last problem you have to say, what two numbers when I add them equal to minus 11, and when I multiply them, is equal to positive 24? So first of all, if I'm multiplying them and I get a positive 24, they're either both positive or they're both negative. Since this is a negative number it tells me that they both have-- I mean they both can't be positive. You can't add two positive numbers and end up with a negative. So we're going to deal with two negative numbers. So essentially, what two negative numbers when I add them equals minus 11 and when I multiplying them equals 24? And hopefully, 8 and 3 pop out at you. Because minus 8 plus minus 3 is equal to minus 11. That's that. And then minus 8 times minus 3 is positive 24. So this can be factored as x minus 8 times x minus 3. And if we look at their choices, they had x minus 3 there. And of course, don't get this confused. If they said, solve this equation, right? This is just an expression. Now it becomes an equation if we put an equal sign. Then we would have this equaling 0 and then you could say the roots of this polynomial or the solutions to the equation are what makes this true. And then the solutions wouldn't be minus 8 and minus 3. The solutions would be what makes this 0. So it's be x is equal to 8 would make this 0, or x is equal to 3 would make that 0. But anyway, I'm going off on a tangent and that's not what they asked you. But I think that confuses people sometimes. OK, 46. Which of the following shows 9t squared plus 12t plus 4 factored completely? OK, now this is an interesting one because immediately, when I look at the numbers, there's not one number that I can just factor out of everything. 9, 12, and 4, they don't have any common factors, so I can't just do that simplification. So we're going to do a little bit more complexity. But the best way to think about is whatever's on the t squared, this is probably kind of-- this whole expression, if we're trying to factor it into two binomials or into one binomial, this is going to be the first term of that binomial squared. So we're going to be dealing with something like 3t. I'm just taking the square root of 9t squared. 3t plus some number. Let's say plus a. Times 3t plus b. And now we can actually just multiply this out and see what happens. Well first of all, they gave us a multiple choice. We could just multiply these out and see what happens. But let's pretend like that they didn't give us choices and we had to do this in a vacuum. If we did this in a vacuum, we would have to factor it ourselves. We wouldn't just be able to test their choices. Let's do that. So if we multiply this out, we have 3t times 3t is 9t squared. Then you 3t times b, so it's plus 3bt. Plus 3at plus ab. So this simplifies to 9t squared plus. Now what are we adding? Let's see. It says 3 times, a plus b, t. So it'd be 3 times, a plus b, t. I just added these two terms and I factored out the t and the 3 plus ab. Well now we can do a little pattern matching, right? We could say a plus b times 3 is equal to 12. So a plus b is equal to what? a plus b is equal to 12. This whole coefficient right here is a 12 up here. So a plus b, it must be equal to 4. Because 4 times 3 is equal to 12. a plus b is equal to 4 and we have a times b is also equal to 4. So the only number that I can think of when I add them I get 4, when I multiply them I get 4 is 2 and 2. Both of these are 2. So if I were to factor this completely I get 3t plus 2 squared, essentially. Because both of these terms are the same thing, and that's choice A. A faster way frankly, to do this might have just been to multiply this out and say, that's the same thing. Anyway, next problem. What is the complete factorization of 32 minus 8z squared? So let's think about this a little bit. So the first thing that I like to do once again is to try to factor out any numbers that are just common to all of the terms. So let me do that. So 32 minus 8z squared. 8 goes into both of these, right? So let's factor out an 8. And actually, let's factor out a negative 8, and I'll show you why did that. Because I like to put the z squared, I like that one to be positive. So let's factor out a negative 8. And so you get a minus 8. And you didn't have to do that. You could have just factored an 8. What's 32 divided by minus 8? That's minus 4. And then minus 8z squared divided by minus 8 is just plus z squared. And so we can rearrange this, so this becomes minus 8, times z squared minus 4. And now I'll review this because this is an Algebra 1 test, so maybe this isn't obvious to you. But in general, if you see something like this, a plus b times a minus b-- and this might be a good exercise for you to multiply this out. But this is equal to a squared minus b squared, because the middle terms cancel out. And that's a good exercise for you to do. So this has that same property. This is a squared minus b squared. This is a perfect square. So then this up here can be factored as minus 8 times-- this is a squared minus b squared. So if we could say a is z and b is 2. Because then when we would have z squared minus 2 squared, which is 4. So if a is z, so it's z plus 2 times z minus 2. This is probably the most common thing you'll see in a lot of factoring examples. You'll see something that has this pattern, a squared minus b squared, and you should easily be able to recognize it. You should be able to prove it to yourself as well that this is the same thing as a plus b times a minus b. Or in this case, z plus 2 times z minus 2. And that is z minus 2. So this is interesting. So I don't see exactly what we wrote. I have a minus 8, but we want a z plus 2 and z minus 2. But they don't have that. But what they do have is this-- let me rewrite it. They don't have any of it, but maybe they have-- they want the minus to be multiplied by one of these other terms. So what we could do is we could multiply the minus because multiplication is associative. It doesn't matter what order you do it in. So we could rewrite this as 8 times z plus 2 times minus 1 times z minus 2. And then this becomes 8 times z plus 2. This isn't minus 1, this is times minus 1. Times what? Minus z plus 2. And now I think that choice is there. Right. So we can rewrite this as 8 times 2 plus z-- I'm just rearranging it-- times 2 minus z. And then that is choice B. And so really, given the choices they gave us, it probably would have been faster to just factor out an 8 from the beginning and not a positive 8. And you would have immediately gotten to 2 plus z times 2 minus z. Anyway, I'll see you in the next video.