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alright we're on problem 44 and they said which is the factored form of 3a squared minus 24 a B plus 48 B squared and if this confuses you with A's and B's instead of an X there I just like to think of you know in this case you know 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 you know 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 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 and let's see if we can factor out a number well share all of these are divisible by 3 right 3 is divisible by 3 2400 by 3 + 48 stew fizzle 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 8 a B right 24 divided by 3 is 8 and there's a minus in front of it plus 48 divided by 60 40 is 3 I'm brain was getting out of me 48 divided by 3 is 16 B 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 variant you to visualize it properly 3 a squared minus 8 B times a plus 16 B squared so if we viewed a squared is 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 okay are there two numbers are there two numbers that add up to minus 8 B and that when I multiply the two numbers I get 16 B squared right so first of 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 this the same number just use twice otherwise I you know it would be weird to get a a squared here so the number that should pop out you say okay what two numbers add up to minus eight 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 4 B is equal to minus 8 B right and minus 4 B squared is equal to 16 B squared so we can factor this out to be 3 let me switch colors three times we said minus 4 B is the number when you add it twice you get minus 8b and you squared you get that so it's a a minus 4 B times a minus 4b and that is choice C next problem 40 I did I skip 45 I did let me copy and paste 45 in here can't skip problems 4045 let me copy and paste it in okay I could just maybe overdo it on this on top of this one let me see if I can erase this look at that alright 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 and that well that's the easiest way or you could test for zeros 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 it's equal to minus 11 and when I multiply them is equal to positive 24 positive 24 so first of all if I'm multiplying when I get a positive 24 that either both positive or they're both negative since this is a negative number it tells me that they're both have I mean they both can't be positive you can't add two positive numbers and how to end up with a negative so we're gonna deal with two negative numbers so essentially what two negative numbers when I add them equals minus 11 and when I multiply them equal 24 and and hopefully eight and three pop out at you because minus 8 plus minus three is equal to minus eleven that's that and then minus eight times minus three is positive 24 so this can be factored as X minus 8 times X minus three and if we look at their choices they had X minus three there and of course don't get this confused if they asked you you know if this was a if they said solve this equation right this is just an expression now it becomes an equation we put equal sign then we would have this equaling zero and then the you could say the roots of this polynomial of the solutions 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 zero so it'd 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 I think that confuses people sometimes ok 46 which of the following shows 90 squared plus 12t plus 4 factored completely ok now this is an interesting one because immediately when I look at when I look at the when I look at the numbers there's not one number that I can just factor out of everything right there's not it's not like you know 9 12 and 4 they don't have any common factor so I can't just do that simplification so we're gonna have to do a little bit more a little bit more complexity but the best way to think about is whatever is on the T squared whatever on the whatever is on the T squared this is probably kind of know this whole expression if we're if we're trying to factor it into two binomials or into one binomial this this is going to be the first term of that binomial squared so we're going to be dealing with something like 3t right I'm just taking the square root of 9 T squared 3t plus some number let's say plus a times 3t plus B and now we could actually just multiply this out and see what happens okay well first of all they gave us a multiple choice we could just really just multiply these out and see what happens but let's pretend like they didn't give us choices and we had to do this in a vacuum if you did this in a vacuum we would have to factor it ourselves we wouldn't just be able to test their choices so let's do that so if we multiply this out we have 3t times 3t is 90 squared then you have 3t times B so it's plus 3b T plus 380 right plus 380 plus ayby plus a B so this simplifies to 9t squared plus now what are we adding the seat says three times a plus B T so it'd be three times a plus B T right I just added these two terms and I factored out the T and the three plus a B well now we can do a little pattern matching right we could say a plus B times three is equal to 12 right a plus B times three is equal to 12 so a plus B is equal to what a plus B is equal to 12 right this whole coefficient right here is a 12 up here so a plus B it must be equal to 4 right 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 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 right both of these are 2 so if I were to factor this completely I get 3t plus 2 squared essentially because they both of these terms are the same thing and that's choice a a faster way frankly if you did to do it this might have just been to multiply this times 0 that's the same thing anyway next problem what is the complete factorization of 32 minus 8 Z 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 8 Z 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 I 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 can adjust factor na but am i what's 32 divided by minus 8 what's minus 4 right and then minus 8 Z 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 this is I'll review this because this is an algebra one 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 right this is a squared minus B squared this is a perfect square so then this up here can be factored as minus eight times eight this is a squared minus B squared so it's going to be eight so if we could say a is Z a is Z and B is two right because then we were to Z squared minus two squared which is four so it's if a is Z so it's Z plus 2 times Z minus 2 this is one of this is probably the most common thing you'll see when in a lot of you know factoring examples you'll see something that has this pattern a squared minus B squared and you should immediately 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 and this is something so I don't see exactly what we wrote right 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 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 you know 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 x 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 at 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 front in the beginning and not a positive 8 and you Raviv immediately gotten to 2 plus Z times 2 minus e anyway I'll see you in the next video