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# The distributive property with variables

CCSS.Math:

## Video transcript

in earlier mathematics that you may have done you probably got familiar with the idea of a factor so for example let me just pick an arbitrary number the number 12 we could say that the number 12 is the product is the product of say 2 & 6 2 times 6 is equal to 12 so because if you take the product of 2 & 6 you get 12 we could say that 2 is a factor of 12 we could also say that 6 is a factor of 12 you take the product of these things you get 12 you could even say that this is this is 12 in factored form people don't really talk that way but you could think of it that way that we we broke 12 into the things that we could use to multiply it and you probably remember from earlier mathematics the notion of prime factorization where you break it up into all of the prime factors so in that case you could break the 6 into a 2 and a 3 and you have 2 times 2 times 3 is equal to 12 and you would say well this would be 12 in prime factored form of the prime factorization of 12 so these are the prime prime factors and so the general idea this like this notion of a factor is things that you can multiply together to get your original thing or if you're talking about factored form you're thinking you're essentially taking the number and you're breaking it up into the things that when you multiply them together you get your original number what we're going to do now is extend this idea into the algebraic domain so if we start with an expression let's say the expression is 2 + 4 X can we break this up into the product of two either numbers or two expressions or the product of a number and an expression well one thing that might jump out at you is we can write this as we could write this as 2 times 1 + 2 X and you can verify if you like that this does indeed equal 2 + 4 X - x we're just going to distribute the 2 2 times 1 is 2 2 times 2 X is equal to 4 X so plus 4 X and so in when we in our in our brains this will often be viewed as or referred to as this expression factored or in a factored form and in this case or sometimes people would say that we have factored out the two but you could just as easily say that you have factored out a 1 plus 2x you've broken this thing up into two of its factors so let's do let's do a couple of examples of this and then we'll think about you know I just told you that hey we could write it this way but how would you actually figure that out so let's do another one let's say that you had I don't know let's say you had six let me just in a different color let's say you had 6 X 6 X plus plus 3 plus no it's just right 6 so 6 X plus 30 that's interesting so one way to think about it is can we break up each of these terms so that they have a common factor well this one over here I can literally 6x literally represents 6 times X and then 30 if I want to break out a 6 30 is divisible by 6 so I can write this as 6 times 5 30 is the same thing as 6 times 5 and when you write it this way you see hey I can factor out a 6 essentially this is the reverse of the distributive property so I am essentially undoing the distributive property taking out the 6 and you are going to end up with so if you take out the 6 you end up with 6 times 6 times so if you take out the 6 here you have an X and you take out the 6 year you have plus 5 so 6x plus 30 if you factor it we could write it as 6 times X plus 5 and you can verify with the distributive property if you distribute this six you get 6x plus 5 times 6 or 6x plus 30 let's do something that's a little bit more interesting where we might want to factor out a fraction so let's say we had the situation let me get a color new color here so let's say we had 1/2 minus 3 halves minus 3 halves X how could we write this in a I guess you could say in a factored form or if we wanted to factor out something and I encourage you to pause the video and try to try to figure it out and I'll give you a hint see if you can factor out one half well so let's write it that way if we're trying to factor out one half we can write this first term as one half times one and this second one we could write as so minus one half times 3x that's what this is three halves X is the same thing as 3x divided by 2 or 1/2 times 3x and then here we could see that we can just factor out the 1/2 and you're going to get one half times one minus 3x another way you could have thought about it is hey look both of these are the our products involving 1/2 an ounce a little bit more confusing when you're when you're dealing with a fraction here but one way to think about it is I can divide out a 1/2 from each of these terms so if I divide out a 1/2 from this 1/2 divided by 1/2 is 1 and if I take out if I take 3 halves and I divided by 1/2 that's going to be 3 and so I took out a 1/2 that's another way to think about it I don't know if that confuses you more or confuses you less but hopefully this gives you the sense of what factoring and expression is and I'll do another example where we even using more abstract things so I could say ax plus ay Y how could we write this in fractured form well these are both both of these terms have products of ay in it so I could write this as a times X plus y and sometimes you'll hear people say you have factored out the a and you can verify it if you multiply this out again if you distribute the a you'd be left with ax plus ay Y