Divide polynomials by x (with remainders)

Simplify the expression 18x to the fourth minus 3x squared plus 6x minus 4, all of that over 6x. So there's a couple ways to think about them. They're all really equivalent. You can really just view this up here as being the exact same thing as 18x to the fourth over 6x plus negative 3x squared over 6x, or you could say minus 3x squared over 6x, plus 6x over 6x, minus 4 over 6x. Now, there's a couple of ways to think about it. One is I just kind of decomposed this numerator up here. If I just had a bunch of stuff, a plus b plus c over d, that's clearly equal to a/d plus b/d plus c/d. Or maybe not so clearly, but hopefully that helps clarify it. Another way to think about it is kind of like you're distributing the division. If I divide a whole expression by something, that's equivalent to dividing each of the terms by that something. The other way to think about it is that we're multiplying this entire expression. So this is the same thing as 1 over 6x times this entire thing, times 18x to the fourth minus 3x squared plus 6x minus 4. And so here, this would just be the straight distributive property to get to this. Whatever seems to make sense for you-- they're are all equivalent. They're all logical, good things to do to simplify this thing. Now, once you have it here, now we just have a bunch of monomials that we're just dividing by 6x. And here, we could just use exponent properties. This first one over here, we can take the coefficients and divide them. 18 divided by 6 is 3. And then you have x to the fourth divided by x to the-- well, they don't tell us. But if it's just an x, that's the same thing as x to the first power. So it's x to the fourth divided by x to the first. That's going to be x to the 4 minus 1 power, or x to the third power. Then we have this coefficient over here, or these coefficients. We have negative 3 divided by 6. So I'm going to do this part next. Negative 3 divided by 6 is negative 1/2. And then you have x squared divided by x. We already know that x is the same thing as x to the first. So that's going to be x to the 2 minus 1 power, which is just 1. Or I could just leave it as an x right there. Then we have these coefficients, 6 divided by 6. Well, that's just 1. So I could just-- well, I'll write it. I could write a 1 here. And let me just write the 1 here, because we said 2 minus 1 is 1. And then x divided by x is x to the first over x to the first. You could view it two ways-- anything divided by anything is just 1. Or you could view it as x to the 1 divided by x to the 1 is going to be x to the 1 minus 1, which is x to the 0, which is also equal to 1. Either way, you knew how to do this before you even learned that exponent property. Because x divided by x is 1, and then assuming x does not equal to 0. And we kind of have to assume x doesn't equal 0 in this whole thing. Otherwise, we would be dividing by 0. And then finally, we have 4 over 6x. And there's a couple of ways to think about it. So the simplest way is negative 4 over 6 is the same thing as negative 2/3. Just simplified that fraction. And we're multiplying that times 1/x. So we can view this 4 times 1/x. Another way to think about it is you could have viewed this 4 as being multiplied by x to the 0 power, and this being x to the first power. And then when you tried to simplify it using your exponent properties, you would have-- well, that would be x to the 0 minus 1 power, which is x to the negative 1 power. So we could have written an x to the negative 1 here, but x to the negative 1 is the exact same thing as 1 over x. And so let's just write our answer completely simplified. So it's going to be 3x to the third minus 1/2 x plus 1-- because this thing right here is just 1-- and then minus 2 times 1 in the numerator over 3 times x in the denominator. And we are done. Or we could write this. Depending on what you consider more simplified, this last term right here could also be written in minus 2/3 x to the negative 1. But if you don't want a negative exponent, you could write it like that.