 So, this is a review of some basic algebraic operations, and once again, here's a disclaimer. This is a review. It assumes that at some point you learn how to handle basic algebraic operations, but maybe you need a little bit of a refresher. It consists of a set of rules for operating with basic algebraic operations, followed by some examples. And again, as a note, this is the worst of all possible ways to learn mathematics. You really, it is an awful way to learn mathematics to memorize a set of rules and follow examples. If you haven't already learned how to handle basic algebraic operations, then this video will not help you. Instead, here's a couple of videos that might prove a little bit more useful. Alright, so one of the important things that you should be already familiar with is the distributive property. If I have real numbers a, b, and c, then the product a times b plus or minus c is going to be a times b plus or minus a times c. It's called distributive property because I've distributed this factor a among both of the sum adds, so a, b plus a, c. And what's important here is some terminology that you'll run into. This allows us to expand a product over on the left-hand side into a sum or difference, depending on whether I have a sum or difference inside. So on the left-hand side I have a product. On the right-hand side I have a sum or difference. If I go this way, I've expanded a product into a sum or difference. On the other hand, if I go the other way, I have a sum or difference over on the left-hand side. Equal says I could replace the one with the other. I have a sum or difference, and I can rewrite it as a product of two things. So if I go from the sum or difference, I can factor, I can rewrite it as a product. So when you hear the word expand, it means you're going to take and apply the distributive property in some fashion. And when you hear the word factor, you're going to use the distributive property going this way. So for example, suppose I want to expand 8x times y plus 7. Well, I can compare the distributive property, a times b plus c, ab plus ac. And if I take a look at what the distributive property is versus what I have, well, I see that a is 8x, b is y, c is 7. So I can expand by mimicking the distributive property here. So I have a times b plus c, that's a times b, 8x times y, plus, same thing as I had inside parentheses, a times c, 8x times 7, 8x times 7 is there. Now I can do a little bit of simplification here. Multiplication is both commutative and associative. There's nothing really wrong with this expression, but it does look a little peculiar. So I'll use the property of multiplication, the associative and commutative property of multiplication to rewrite this. Commutativity says I can rewrite this in any order. This is 8x times 7 is the same as 7 times 8x. Associativity says I can multiply 7 times 8 first, then multiply by x. And I do know how to multiply 7 times 8, gives me 56, and there's my expansion. Now along with the distributive property, one thing we also use frequently in conjunction with it is this notion of combining like terms. And so two terms are said to be like terms if they have the same variables raised to the same powers. And this is an absolute requirement. You have to have all the variables be the same and all the powers on those variables be the same. So here's 8x to power 5 thirds, y to the third, 32 to the fifth x to the 5 thirds, y to the third. And I have variables x and y, I have variables x and y, x is raised to the 5 thirds, x is raised to the 5 thirds, y is raised to the third, y is raised to the third. So these are like terms. On the other hand, let's take something like 4xy and 4x. These are unlike terms. The variables are different. This has xy, this has x in it, and the variables are different. So they can't be like terms. And importantly, even if they have the same variables, the exponents must also be the same. So here I have an xy, here I also have an xy. The variables are the same, but here x is raised to the second power and here x is not. So the terms here are unlike terms. If the terms are like, if I have like terms, I can add the coefficients and here's the important thing. We do not want to change the variable part. So here with the like terms, I keep the variable portion, x to the 5 thirds, y cubed, but I add 8 and 32 to the fifth. So for example, let's say I want to simplify this thing. A little bit of analysis goes a long way. This is something that says parentheses. Do stuff first, y plus 7. Well, the terms are unlike. This has a y variable and this does not. The terms are unlike. There's nothing else I can do with that set of expression inside the parentheses. Now, order of operation says the next thing is I take care of the multiplication. This times this. And well, this is a product, so I can use the distributive property. So I'll expand that out. This is 8x times y plus 7. Well, we already did that. That's 8xy plus 56x. And then the 4xy doesn't change. Here's an important rule. Whenever you do mathematics, paper is cheap. Understanding is priceless. And here, I've done something with this 8x times y plus 7. I have not done anything with the 4xy, so I need to carry that along. A lot of success in mathematics is bookkeeping. I haven't done anything with the 4x plus y. It's still there. Well, now I have a sum. So I have, looking at this, well, this is a not like term because this does not have the y variable. Both of these terms have x. They have y. And the variables are raised to the same power. x is raised to the first power. x is raised to the first power. y is raised to the first power. y is raised to the first power. So these are like terms. And so I can combine them by adding the coefficients without changing the variable parts. So I'll add the coefficients 8 and 4. I'll keep the variable part xy, 8 plus 4 times xy. That's going to be 12xy plus 56. And just as a side note, you might notice that our ability to combine like terms relies on the distributive property. So there's my simplification. One more example here. Sometimes when we have a subtraction, things are somewhat more complex. We have to be a little bit careful with that. So here, remember that the rules of integer says that if I have a subtraction a minus b, it's the same as a plus negative b. So I should rewrite the subtraction just to make this a little easier to work with. So instead of 12x minus 3x minus 4, this is 12x plus negative 3 times x minus 4. So I can use the distributive factor. There's a product here, negative 3 times something I can distribute over. So I'll distribute that negative 3 in. That's negative 3 times x minus. I keep the same sign here, negative 3 times 4. I can use the properties of integer arithmetic to rewrite this. This is plus a negative. So that's the same as minus. This is minus a negative. So that's the same as a plus. So that's my plus, my negative. That's minus 3x. This is minus 3 times 4, I should say. Minus 3 times 4 is negative 12. And now I have minus a negative. So I can add. And I do have like terms here, 12x, 3x, same variable, same power. So I can combine the like terms. I can add those coefficients, or in this case subtract 12. Minus 3 is 9. Keeps the variable part 12. Nothing happens to, so that has to get carried down. And there's my simplification.