Hi, and welcome. My name is Anthony Varela, and today we're going to divide polynomials by monomials. So we're going to talk about algebraic factoring, we're going to divide algebraic terms, and we're going to look at examples that have remainders and examples that do not have remainders.
So first, let's talk about algebraic factoring. So if we wanted to factor 12x minus 3, what we have to do is look at each of our terms, 12x and negative 3, and identify some common terms or common factors between these two terms. So I'm going to write out, then, the common factor-- or write out all factors of 12x and all factors of 3, and we'll see what they have in common.
So I'm going to write up 12x as 2 times 2 times 3 times x. So I'm breaking it down into all of its prime numbers. So 2 times 2 times 3 is 12, and then we have one factor of x. Well then taking a look at 3 now, 3 is a prime number, so it's just that number and 1, so 3 times 1. So looking at factors that they have in common, that would be just 3. So I can factor out a 3 from 12x minus 3.
So here is my factor that I'm factoring out. And then in parentheses, I would just have 12x divided by 3 and negative 3 divided by 3. So 12x divided by 3, the common factor, is 4x. And then I have my minus 3. And when I divide that by 3, I just have minus 1. So 12x minus 3 is the same as 3 times 4x minus 1. We could check with distribution. So 3 times 4x gives me 12x, and 3 times negative 1 gives me negative 3.
Let's take a look at another example. Here we have 8x cubed minus 6x. So we're going to list out, then, all the factors that make up 8x cubed. So 8 can be broken down into 2 times 2 times 2, and then x cubed is x times x times x. What about 6x? Well, I can break 6 down into 2 times 3, and then we just have one factor of x.
So now take a look, then, at the common factors. Well they share a 2 and they share an x. So the greatest common factor between 8x cubed and 6x is 2x. So that's going to be your outside factor that we factor out. And another way to figure out, then, what goes in the parentheses is just multiply everything else that we didn't highlight.
So 2 times 2 is 4, and then x times x is x squared. So I'm going to have 4x squared, and then here I just have 3. And it's a minus 3 because these are separated by subtraction. So 2x times 4x squared minus 3 is the same as 8x cubed minus 6x. We can check this with distribution. 2x times 4x squared is 8x cubed. 2x times negative 3 is minus 6.
So how does this then relate to dividing a polynomial? Let's take a look at 18x cubed minus 15x squared plus 9x, so that's our polynomial. And we're going to divide this by 3x. Now one thing that I like to do is separate this into individual fractions. So we're going to have 18x cubed divided by 3x, and then we'll subtract 15x squared divided by 3x, and then we'll add 9x divided by 3x.
And now we're looking at our coefficients that we see, and we'll just divide those, and then we'll divide our variables. So looking at 18 divided by 3, well that's going to be 6. And then x cubed divided by x is x squared. So I've simplified 18x cubed over 3x as 6x squared. So we're canceling all the common factors. This includes our variables.
So how about our next term? We have a minus 15x squared divided by 3x. Well 15 divided by 3 is 5, and then x squared divided by x is x. We have a minus 5x here. And finally, 9 divided by 3 gives me 3, and x divided by x is 1. So I don't have to write that at all. I can just say, plus 3. So 6x squared minus 5x plus 3 is this polynomial here divided by 3x. It worked out without a remainder because 3x was a factor of everything here.
Let's take a look at an example that will have a remainder then. So notice I just tacked on a plus 6. So all of this is the same, except of our plus 6 but we're dividing all of this by 3x. So we already know what our first couple of terms will look like from our example before. We're going to have 6x squared, then we're going to have minus 5x, and then we're going to have plus 3. But now we have to divide 6 by 3x. So what does that look like? 6 divided by 3x.
Well this doesn't divide evenly. That means that they certainly share a factor-- a numeric factor-- but not an algebraic factor. So how are we going to simplify this? Well we can certainly simplify 6 divided by 3, so 6 divided by 3 is going to be 2. But we still have that x in the denominator. So we can still simplify this. We have 6x squared minus 5x plus 3, and then we have plus 2 over x. We were able to simply 6 divided by 3, but we still have an x factor of x in our denominator. So this, then, would be a remainder, the remainder of 2 over x.
So let's review polynomials divided by monomials. Well with factoring, we're looking for common factors between terms, and we would cancel them out or factor them out. And we can check this with distribution. If we distribute, we should get back to our original expression.
When you're dividing polynomials by monomials, you want to cancel all common factors, and this includes your variables. And sometimes you'll have a remainder when it doesn't divide evenly. So it's going to look like a fraction, and you may be able to simplify that fraction. So thanks for watching this tutorial on polynomials divided by monomials. Hope to see you next time.