Hi. This is Anthony Barrilla. And today we're going to be adding and subtracting polynomials. So we're going to talk about how to identify like terms, and then we'll be combining like terms. And this is all going to aid us then when we're adding and subtracting polynomials.
So, first, we're going to talk about like terms and how to combine them. Well, first, we need to identify what like terms are. And we're looking for the same variables and the same variable powers. So if we have this expression, x plus 3 minus 2x minus 4, we see that we have an x term, a constant, another x term, and another constant.
So our like terms, then, we're looking for x terms. And then we have constant terms. So looking at our x terms, we have x and negative 2x. And with our constant terms, we have 3 and negative 4. Now, once we have our like terms identified, we can combine them by adding their coefficients. So to combine our x terms, we have an implied coefficient of 1. That's always going to be the coefficient if it's not actually written there. And so we're going to then add that to the coefficient of the other x term, negative 2x.
So we have 1 plus negative 2. We can think of that as subtraction, if you're adding a negative. So 1 minus 2 equals negative 1. So that's going to be the coefficient of our combined x terms. For our constant terms, we have 3 minus 4. And once again, if you're adding a negative number, you can think of it as subtracting a positive. So our constant, then, when combined is also negative 1.
So now, how do we interpret then a simplified expression now that we've combined our like terms? Well, this is the coefficient of our x term. And then this is our constant. So simplified, we'd have negative 1x. So I can just write negative x. I don't have to write the 1. And I have minus 1. So let's practice identifying like terms with a more complicated expression. So we have more variables and we have more powers. So looking at our first term, that's 2x cubed. So we have an x cubed term.
Let's look at our next term. We have x squared y. So that is one of our terms. Then we have a minus 2xy squared, and we might be thinking, OK, well, I see an x, I see a y, and I see a power of 2. But this is a bit tricky. Our power of 2 is still associated with y in this term, whereas it was associated with x over here. So this is x times x times y. This is x times y times y. So it's a different kind of term. So we also have an xy squared term.
Well, then we have another x cube term. We already have that listed. And then we have another x squared y term. We already have that listed. So now that we have identified what kind of terms we have, let's go ahead and sort then this expression. So for our x cubed terms, we have 2x cubed and 1x cubed. For our x squared y terms, we have x squared y and minus 3x squared y, so we're bringing that sign in front down with it here. And with our xy squared terms, we just have the negative 2xy squared.
So now we can combine these like terms by adding their coefficients. So I'm putting in my implied coefficients here, our 1's. And then we're just focusing on adding then those coefficients. So 2x cubed plus 1x cubed is 3x cubed. 1x squared y plus negative 3x squared y would then be a minus 2x squared y. And then we just have our negative 2xy squared term here.
So simplifying this messy polynomial up here, we can just bring all these together into one expression. So now, how does this help us then add and subtract polynomials? So let's take a look at this polynomial addition problem. So here are two polynomials that we're going to add together. And one thing that I like to do when adding and subtracting polynomials is to align my terms vertically. And I'm going to put these in order too.
So this is what I mean by aligning them vertically. So I'm taking my first polynomial and I have it in order of descending degrees, so degree 3, degree 2, degree 1, and then my constant term. Now, I'm going to take this polynomial here, which we'd like to add, and I'm writing it aligning our terms vertically. So here I have my x cubed term. Here we have an x squared term.
Now, this polynomial doesn't have an x term. So I'm leaving space there so that my constant terms can be aligned vertically. So now I'm going to write in our implied coefficients. So when we have a term that is missing, so to speak-- so here we don't have an x term-- we're going to write a coefficient of 0. And when we don't see a coefficient in front of a term, it's implied to be 1. So I'm adding my coefficients of 0 and 1.
So now I can focus just on adding those coefficients, and then I'll just bring down the variable and the variable power. So let's start over here. We're going to add 2 plus 1. So it will give us 3x cubed. Next we have 1 plus negative 4. So I can think about that as 1 minus 4. So I have a minus 3x squared. Now I have a negative 2 plus 0. So that's going to be a negative 2x. And then adding my constant terms, negative 4 plus 8, that gives me a positive 4.
So here then is my polynomial sum. Let's go through an example with subtraction. So here are the two polynomials that I'd like to subtract. And here, we're following the same process. We're just going to be subtracting the coefficients rather than adding them. So let's go ahead and align these vertically. And I'm going to remember to put in my implied coefficients.
So this polynomial does not have an x term as this one does. So I have 3x squared plus 0x plus 8. This then leaves enough space for me to align my x squared terms, align my x terms, and align my constant terms. So now I'm going to subtract these coefficients. So 3 minus 2 is 1. And then I have x squared. Notice you don't have to write the 1 in your final solution. Next, we have 0 minus 4. So that's going to be a minus 4x. And then we have 8 minus a negative 2. And when you're subtracting a negative, you can think of that as adding a positive. So this would be 8 plus 2 or positive 10. So there then is my polynomial difference.
So let's review adding and subtracting polynomials. Well, we talked about like terms, having the same variables, and same variable powers. And we can combine like terms by adding the coefficients. So when we're adding and subtracting polynomials, you want to align your terms vertically. And it helps to put them in order-- remember that's descending degree-- and use those implied coefficients. Use a coefficient of 0 when a term is absent, and use a coefficient of 1 when that actual coefficient in front of a variable is absent, it's implied to be 1.
So thanks for watching this tutorial in adding and subtracting polynomials. Hope to see you next time.