The distributive property & equivalent expressions
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- 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 of say two and six; two times six is equal to 12. So because if you take the product of two and six, you get 12, we could say that two is a factor of 12, we could also say that six is a factor of 12. You take the product of these things and you get 12! You could even say that this is 12 in factored form. People don't really talk that way but you could think of it that way. We broke 12 into the things that we could use to multiply. 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 six into a two and a three, and you have two times two times three is equal to 12. And you'd say, "Well, this would be 12 "in prime factored form or the prime factorization of 12," so these are the prime factors. And so the general idea, 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 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 two plus four 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 two times one plus two X. And you can verify if you like that this does indeed equal two plus four X. We're just going to distribute the two. Two times one is two, two times two X is equal to four X, so plus four X. So in our algebra brains, this will often be reviewed as or referred to as this expression factored or in a factored form. Sometimes people would say that we have factored out the two. You could just as easily say that you have factored out a one plus two X. You have broken this thing up into two of its factors. So let's do a couple of examples of this and then we'll think about, you know, I just told you that we could write it this way but how do 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 six X six X plus three, no, let's write it six 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, six X literally represents six times X, and then 30, if I want to break out a six, 30 is divisible by six, so I could write this as six times five, 30 is the same thing as six times five. And when you write it this way, you see, "Hey, I can factor out a six!" Essentially, this is the reverse of the distributive property! So I'm essentially undoing the distributive property, taking out the six, and you are going to end up with, so if you take out the six, you end up with six times, so if you take out the six here, you have an X, and you take out the six here, you have plus five. So six X plus 30, if you factor it, we could write it as six times X plus five. And you can verify with the distributive property. If you distribute this six, you get six X + five times six or six X + 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 new color here. So let's say we had 1/2 minus 3/2, minus 3/2 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? I encourage you to pause the video and try to figure it out, and I'll give you a hint. See if you can factor out 1/2. Let's write it that way. If we're trying to factor out 1/2, we can write this first term as 1/2 times one and this second one we could write as minus 1/2 times three X. That's what this is, 3/2 X is the same thing as three X divided by two or 1/2 times three X. And then here we can see that we can just factor out the 1/2 and you're going to get 1/2 times one minus three X. Another way you could have thought about it is, "Hey, look, both of these are products "involving 1/2," and that's a little bit more confusing 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 one. And if I take 3/2 and divide it by 1/2, that's going to be three, 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 it confuses you less, but hopefully this gives you the sense of what factoring an expression is. I'll do another example, where we're even using more abstract things, so I could say, "AX plus AY." How could we write this in factored form? Well, both of these terms have products of A 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.