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Terms and Factors in Algebraic Expressions

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Today we're going to talk about terms and factors in algebraic expressions. So we'll start by going over what exactly terms and factors are. We'll go over some language used when talking about terms and factors. And then we'll do some examples, adding and subtracting terms by using something called combining like terms.

So let's start by looking at this algebraic expression 4x to the third. 4x to the third is just one term. Now a term is just a collection of numbers, variables, and powers that are being multiplied together.

So let's go over some language that we use to talk about a term. So the first thing is the number that's in front, we call that the coefficient. And a coefficient is just the number that's in front of the term being multiplied by the variable.

So the next thing that we have is our variable. And the variable is just a quantity that can change, usually expressed as a letter but could also be a symbol.

And the last thing is our power or our exponent.

So the other thing to notice is that 4 and x to the third are both factors of our term because they are being combined together through multiplication.

Let's look at a few different types of algebraic expressions. So we started by looking at in our last example an expression with just one term and that you can call a monomial. So in this monomial, 3 is our coefficient, y is our variable, and 5 is our power.

We can also have an expression with more than two terms, and you can call an expression with two terms a binomial. So in this expression, our first term has a coefficient of negative 4, the variable is m, and the exponent is 2.

And then it has a number that's by itself. And when you have a number with no variable component or a term with no variable component, we call that a constant. And that's because if it has no variable, then it's not changing. It's just always going to be equal to 8.

And we also have an example of an algebraic expression with three terms. You can call an expression with three terms a polynomial. So in this polynomial, our first term has a coefficient of 6, the variable is x, and the exponent is 3. Here, our coefficient is 2, the variable is still x, and our exponent is 2. And in our last term, our coefficient is 5, our variable is x, and even though it looks like we don't have an exponent, you can think of there as being an exponent of 1 because x by itself is the same as x to the first power.

So when we refer to terms, we typically look at them in terms of their variable and their exponent. And that's because that's how we can know if two terms are like terms. They need to have the same variable and the same exponent. So let's use that idea to look a little bit further on how you can combine like terms.

So let's do some examples combining like terms. Remember, we talk about terms in terms of their variable and their exponent. So when we are combining terms, we know that we can combine them if they have the same variable and the same exponent.

So for my first example, I see that I have a 5y and a negative 3y. Since they both have the same variable, y, and they both have the same exponent-- which is actually a 1, we just normally don't put an exponent of 1-- I know that I can combine these two terms. And I do that by adding or subtracting their coefficients. So I've got 5 and a negative 3, which would give me 2. So 5y and a negative 3y is going to give me 2y.

Then I've got negative 6x to the third and a 5y to the third. So they both have the same exponent or power, but they have different variables. So these two terms are not like terms. I can't combine them. So instead since I can't combine them, I'm just going to bring them down. And since I can't combine any other terms, this is as simplified as my original expression can get.

For our second example, if you're feeling pretty confident, go ahead and pause and check back with us later and see how you did.

So here I see that I have a negative 3m to the fourth and a negative 8m to the fourth. Same variables, same exponents, so I can combine them by adding or subtracting my coefficients. So a negative 3 and a negative 8 is going to give me negative 11. And I'll have m to the fourth.

Then I see I've got 2n to the seventh and negative 6 n to the seventh. Since both of these terms have the same variable and the same power or exponent, I know that we can combine them. So positive 2 and a negative 6 is going to give me a negative 4n to the seventh.

These have different variables and different exponents, so I know that I can't combine them. So this is as simplified as my original expression can get.

Let's go over our key points from today. Make sure that you get them in your notes if you don't have them already so you can refer to them later.

So we started by defining a term as a collection of numbers, variables, and powers. And we referred to terms by their variable and their power or exponent. Like terms are terms that you can combine together with adding and subtracting, have the same variable and the same power. And then we talked about a term that has no variable is called a constant. And that's because without a variable, it's not changing. It is constantly the same number.

So I hope that these key points in the examples that we did helped you understand a little bit more about terms and factors in algebraic expressions. Keep on practicing, keep using your notes, and soon you'll be a pro. Thanks for watching.