Hi, and welcome. My name is Anthony Varela, and today we're going to multiply monomials and binomials. Now, while monomials only have one term and binomials only have two terms, there can be lots of different ways that this multiplication can look depending on our variables, our coefficients, and our variable powers.
So we're going to take a look at an example of distribution with just a single variable, then we're going to take a look at multiplying two variables, so we'll get into some multi-variable expressions. And then we're going to combine these ideas and see what happens when we have to distribute with multiple variables.
So let's talk about distribution. Here is an example of multiplying a monomial and a binomial that would require distribution. So we have 3x multiplied by 3 plus 2x. And with distribution, we're just going to multiply this outside factor by every term inside.
So first we'll multiply 3x times 3. So this just gives us 9x. We're multiplying 3 times 3, and then we have an x variable here. Now when multiplying, then, 3x times 2x, we get 6x squared. So let's think about what happened here. Well, we multiplied those coefficients, 3 times 2, to get positive 6, but we have the same kind of variable, so in this case, they're both x's, we add their variable powers. This is one power of x, this is one power of x, so we have two powers of x here. 6x squared.
Now when we're writing out our polynomials, we prefer to order our terms from the highest degree to the lowest degree. So while this isn't necessarily incorrect, it's just not a preferred way to write the product. So I'm actually going to switch this around so we have our x squared term first-- 6x squared plus 9x.
Let's take a look at another example. A little bit more complicated, but we still have the monomial negative x squared. And we're going to multiply that by the binomial, 2x squared minus 4x. So the first step is to multiply negative x squared by two x squared. So multiplying the coefficients, this is a negative 1 times 2, so I get negative 2. And we can add the variable powers here. Two powers of x and two powers of x, so we have four powers of x.
Now we're going to distribute negative x squared into minus 4x. So we have a coefficient of negative 1 multiplied by negative 4. So we have positive 4x cubed because we're combining two powers of x with one power of x. So our product is negative 2x to the fourth plus 4x cubed.
Well now let's introduce multiple variables. So how would we multiply 4x and 3y? Well we're still going to be multiplying our coefficients, so we're going to have a coefficient of 12. That's 4 times 3. And now, because x and y are not the same variable, we would just then write xy.
Now another note about how we prefer to order things. So we order them from highest degree to lowest degree of the terms, but what we also like to do in individual terms is list those variables alphabetically. So all that means is that we'll more often than not see 12xy rather than 12yx because we like to write our variables in alphabetical order.
So let's take a look at another example. First step, we're going to multiply those coefficients. 2 times negative 8, so we get negative 16. And now, because we are dealing with different variables, we're just going to write x squared, y cubed. It's as simple as that.
So now let's put distribution back into the mix here-- distribution with multiple variables. And this is the expression that we'd like to distribute. 3y squared multiplied by y minus 2xy. So the first step is to multiply 3y squared by y. This is going to give us 3y cubed. Multiplying our coefficients of 3 by the implied 1 to get a coefficient of 3 and adding those variable powers, 2 plus the implied 1 to get y cubed.
Next, we're going to multiply 3y squared by minus 2xy. So how would this look? Well, we have a coefficient, 3 times negative 2, so that's a minus 6. And here, we have two powers of y and one power of y, so that's how we get y cubed. And we have zero powers of x and one power of x here, so that's how we get our 1 power of x here, listing our variables alphabetically.
Well now let's take a look at their degree. Well 3y cubed has a degree of 3, but because we add all of the variable powers, negative 6xy cubed actually has a degree of 4. So we'd like to write this term first, and remember to carry over that subtraction. So this is negative 6xy cubed plus 3y cubed.
So let's review multiplying monomials and binomials. We talked about distribution as multiplying the numbers that we see, so those coefficients. And in some cases, we can add those variable powers, like when we have just one variable. So 3x times x squared would be 3x cubed. Here is an example that we walked through where we had more than one variable, so we didn't always simply add those variable powers.
When ordering polynomials, we order from highest degree to lowest degree, and we also like to list our variables with individual terms in alphabetical order. And finally, we talked about the degree of a term as being the sum of all variable powers. That's important to properly order your terms by degree.
So keep in mind that if we have more than one variable in a single term, you're going to have to add those variable powers. It could be tricky sometimes. Here has a degree of 3, even though we see an exponent of 2. That's because we have an implied exponent of 1 here. And this term has a degree of 2, even though we don't actually see the number 2. This is an implied exponent of 1, and this is an implied exponent of 1. So thanks for watching this tutorial on multiplying monomials and binomials. Hope to see you next time.