I copied and pasted the next — because they had little bit of a drawing, so I thought it's better than me drawing it myself. From problem number thirteen it says—and I cut it off a little bit, what is the volume of the figure below? Well the volume is just the width times the height times the depth. So the volume is just going to be-- we could say the volume is going to be, the width is (x+4)*(x+1)*(x+6) right? (x+4)*(x+1), we can do this in our head but I always-- well let's just — I always find it easier to do it this way (x+4)*(x+1); 1*4 is 4, 1*x is x, and then you have the x place; x*4 is 4x, x*x is x2, fair enough? So that equals x2+5x+4 and now we have to multiply that times (x+6) right? We did (x+4) and (x+1) this was the 6, (x+6) that is this side.
So now we multiply this, times (x+6); 6*4 is 24, 6*5x is 30x, 6*x2 is 6x2. I hope you can read what I'm writing, scroll down a little bit and now we do the x’s place. I will do it in a different color because I think I’m going to run out of —x*4 is 4x, x*5x is 5x2; — x*x2 is x3, to run into the white area. So this is x3; what is 6+5? Plus 11x2 plus — 34x plus 24, so — all of this race to x3 plus 11x2 so it’s either B or D plus 34x that is that one, plus 34x plus 24 so the choice is B.
Next problem: the next problem, All right; 8a3+c3 which of these choices? So that it’s interesting, this is not a trivial thing to just factor you know — we will have to do some intuition here. So let’s think about it, in order to have a positive number for the a term; every time we are multiplying it, — we have to have a positive so this is 2a*2a*2a that would get us to 8a3 but there is something interesting about choice A. Why? I already feel like canceling out choice A. Choice A is essentially (2a+c)3 right? Choice A is 2a+c to the third power. And every time you multiply this out; every time you multiply 2a+c times itself, you are going to add more terms, no terms are going to cancel out, I mean if you were to square this, you are going to get you know just to show you get 4a2+4ac+c2 and then to get the third part you would multiply that times 2a+c so you are going to get this thing with you are going to get four terms. So this — choice A is definitely not the case.
In order for one of these to simplify to this we are going to have some positive and negatives there that cancel out terms as you expand these polynomials. So let’s look at the other choices, see if they make any sense, 2a times and I like looking at the a terms and what generates the c3 terms and maybe we could cancel out some things. 2a*4a2 yeah that is 8a3, but –c*c2 that is –c3 right? So when I don’t even have to worry about multiplying this whole thing out. I know because the c3 term is going to come only from this term multiplied by this term. All right and there will be other terms that hopefully will cancel out but we know that choice B cannot be the choice because –c*c2 is –c3 and this is a plus right here. So we know B cannot be the right answer.
Let’s see, choice C; I can use the same logic we have a –c and we have a +c and we have a +c here, we would have to have two plus c’s in order for this to work so this cannot be the answer. So just by deductive reasoning I can say it’s probably choice D but I’m going to prove it to you since we are here to learn not to just answer things correctly. So if I were to multiply those two numbers I have; (4a2-2ac+c2) and I’m multiplying that times (2a+c) —. So I’m going to mix up all of the terms a little bit but you will get the idea so, the important thing when you are multiplying you can do this in your head is that each of these terms times each of those so: c times all of that plus this times all of that, so c times all of that is c*c2 and I’m not doing it the right place notation like I did last time because it gets a little more confusing now. But c*c2 is c3, c*-2ac is -2ac2 right? c*4a2 is 4a2c and then now we can do this one 2a*c2 that is the same thing as, I’m going to put it in this place because it turns into ac2 right? ac2 and I have a two, so 2a*c2 is 2ac2, 2a*-2ac that is if you get a -4a2c so that is -4a2c this is a plus then 2a*4a2 is 8a3 and luckily enough we see we get 8a3 — this cancels out, you get +c3 so we’re right by doing our deductive reasoning, the choice D was the answer.
Next problem, let me uh we have run out of space, run out space. Let me see what the next problem even is. Okay, let me see where was I? Okay I see, okay problem 15; the total area of a rectangle is 4x4-9y2, which factors could represent the length times the width? Okay so essentially they — are saying the length times the width is going to be equal to the area which is 4x4-9y2. So to some degree we have to factor this into multiple expressions and the first thing you should always look at when you have to factor any type of an expression especially if there is only two terms in the expression. Does it fit the form a2-b2? Because if it does, you can factor this into (a+b)*(a-b) and if you don’t trust these equations that I have made multiple videos about it but just multiply them; (a+b)*(a-b)=a2-b2 and luckily this fits that format right? If a lets see if a2 was equal to — 4x4 then what is a? a would be equal to 2x2 and if b2 — is equal to 9y2 then b is equal to the square root of that which is 3y, right? And so this definitely is equal to a2-b2 so this expression right here can be factored into (a+b)*(a-b) so (a+b) is (2x2+3y)*(a-b), (2x2-3y), Lets see which choice is that? 2x2+3y, okay this was the order choice A says (2x2-3y)*(2x2+3y) which is the same thing you can switch orders when you are multiplying so the answer is A.
Next problem, problem 16, I will switch colors arbitrarily, problem 16; which product of factors is equivalent to, let me copy and paste this because this is an interesting so let me copy and paste this problem, okay I have copied it and now let me paste it here, okay which product of factors is equivalent to that thing over there? So let’s simplify that, that is my first impulse, oh no, no, no, we don’t even have to simplify it, we don’t even have to simplify it, this fits our same pattern right? If we can fit the pattern (a2-b2) so look at this thing, this has (x+1)2-y2 so if we said that a is equal to x+1 and b is equal to y then we’d the pattern, right? This is a2-b2 so then this thing can just factor into (a+b)*(a-b), — so what is (a+b)? It’s x+1+y right? That is a+b, and then (a-b) is? x+1-y and that is choice D. That problem was easier than I thought it was going to be, choice D right there — all right and I’m out of time again I will see you in the next video.