 So one important skill is the ability to add, subtract, and later on multiply and divide algebraic terms. Before we discuss the arithmetic of algebraic terms, let's talk about the terms of algebraic terms. Definitions are the whole of mathematics, all else is commentary. And so we might begin with the following. A term is a product of some numbers, the coefficient of the term, and possibly some variables. The degree of the term is the sum of the powers of the variables. We'll also define like terms. Two terms are like if they have the same variables to the same powers. It's useful to keep two important ideas in mind. First of all, x to the zero is always equal to one, provided x is not actually equal to zero. And we can always include a factor of one in any term. x is equal to one times x. And what this means is that even if we don't have a number written down as a coefficient, there's an implied one that's always being multiplied. And even if we don't have any variables at all in our term, there's an implied x to the zero in every term. For example, let's take a look at an expression like this and see if we can find the coefficients and degrees of the terms. And while we're at it, let's see if we can identify any like terms. So a term is a product of some numbers and possibly some variables. So this thing here, three x squared y cubed is a term. It's a product of some numbers, three, and some variables, x squared y cubed. The coefficient is the number that's being multiplied. So here the number three is being multiplied. The degree of the term is the sum of the powers on the variables. So that's x to the second, y to the third. So the degree is two plus three or five. This next thing, eight x, y, again it's a product of some numbers and some variables. So it's a term. Its coefficient is the number eight. And its degree, both of the variables x and y are raised to the first power. So the degree will be one plus one or two. This next thing's a little peculiar because it's not actually a term. Five is a number, but x plus seven is not a variable. It's a variable plus something else. So five times x plus seven is not a term. The next thing, to square root seven pi to the seven x squared y cubed, is a product of some numbers and some variables. So it's a term. The coefficient is the number part to square root seven pi to the seventh. Remember pi is itself a real number. And the variables are to power two and three, so the degree is two plus three or five. And it's worth noting that three x squared y cubed and two root seven pi to the seventh x squared y cubed have the same variables, x and y, raised to the same powers, two and three. And this means they are like terms. The importance of identifying the like terms is that we can combine the like terms using the distributive property. Remember that for any real numbers A, B, and C, A times the quantity B plus or minus C is AB plus or minus AC. And because multiplication is commutative among the real numbers, that's also true for B plus or minus C times A. So, for example, 8xy plus 4xy, quick check, they both have the same variables, x and y, and they're raised to the same powers, both one. So, 8 times x plus y plus 4 times x plus y, my distributive property says that I can split off the x, y, and leave the other terms inside the parentheses, 8 plus 4. But I know what 8 plus 4 is. That's going to be 12. And so when I add these two expressions, I get 12xy. And this suggests a very useful thing when adding like terms add the coefficients, the variables remain unchanged. Now, it's worth pointing out that while the distributive property holds for both addition and subtraction, additions have two very useful features that mean that we'll always, wherever possible, want to write things in terms of addition. First of all, an addition could be rearranged in any order that we want. And second, if we only have additions, we don't have to worry about the parentheses, we can just drop the parentheses. So, A plus B plus C, well, that's just A plus B plus C. So, let's take a look at 4x plus 7y plus 3x plus y. So, while we don't need to do it, we can rearrange the addition in any order that we want to. So, let's rewrite this so our like terms are at least close to each other. When we add like terms, we want to add the coefficients. So, 4x plus 3x will be 4 plus 3, that's 7, and the variable remains unchanged, x. 7y and y are like terms, but there's a problem. Y doesn't appear to have a coefficient. Well, actually it does. Remember, we can always include a factor of 1 in any term. So, x is equal to 1 times x, y is equal to 1 times y. And so, now my y terms have coefficients 7 and 1, and I'll add them together to get my coefficient of the sum 8. And so, my sum is going to be 7x plus 8y.