 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 discuss the terms of algebraic terms. We define a monomial term is the product of some numbers, the coefficient of the term, and possibly some variables raised to whole number powers. The degree of the term is the sum of the powers of the variables. Now, in a perfect universe, we would always speak exactly, and we would not never speak using bad English. Unfortunately, we don't live in that universe. And we often omit the word monomial and just talk about the terms of an expression. And so it's useful to remember the word term usually refers to monomial terms, except when it doesn't. So let's find the coefficients and degrees of the terms in this expression. So remember when we say term, we usually mean monomial term. Definitions are the whole of mathematics, so let's pull in our definition of terms, coefficients, and degrees. And let's look for some terms. So this first thing, 5x to the third, is a term because it's the product of some numbers, 5, and some variables, x to the third. And so we can talk about the term 5x to the third, with coefficient 5, the number, and degree 3, the exponent on our variable. What about the next thing, x to the second? Here it's helpful to remember we can always put a factor of 1 in front of any expression. So x squared can be rewritten 1x squared, and now it's the product of a number and some variables, and so it's a term. Its coefficient is 1, the number part, and its degree is 2, the power on the variable. The next thing is the product of a number and some variables, so we have the term 3x to the fourth y. And here another useful rule of exponents is worth remembering, for any x, x to power 1 is equal to x. And so that means the term 3x to the fourth y can be rewritten as 3x to the fourth y to the first. So it will be a term with coefficient 3 and degree the sum of the exponents for plus 1 or 5. The next thing is 5. Here another rule of exponents is helpful to remember, for any x not equal to 0, x to power 0 equals 1. So this means 5 can be rewritten as 5x to the 0. And since 0 is a whole number, this is now a product of a number and some variables raised to whole number powers. And that means that 5 by itself is also a term. The coefficient is the number, 5, and the degree is the exponent, 0. And this brings up a useful definition. 5 is a constant, it has no variable parts to it, and so we define a constant term to be a term with a degree of 0. And finally this last thing in our sum, 5 times x plus 7. Well, this is not the product of a number and some variables, so this is not a term. Remember that in this context, when we talk about a term, we're really looking at a monomial term. We say that two terms are like if they have the same variables to the same powers. And we can combine like terms using the distributive property. We'll go into details in a moment, but the thing to notice here is that using the distributive property, we'll take two like terms and replace them with one term. And this reduces the number of like terms. Which means that if we keep combining like terms, we will eventually run out. And this leads to the following. We say that an expression is simplified if it has no like terms. 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 1. So, 8 times x plus y plus 4 times x plus y, my distributive property says that I can split off the xy 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.