Divide polynomials by monomials (with remainders)

The quotient of two polynomials-- a of x and b of x-- can be written in the form a of x over b of x is equal to q of x plus r of x over b of x-- where q of x and r of x are polynomials and the degree of r of x is less than the degree of b of x. Write the quotient 7x to the sixth plus x to the third plus 2x plus 1 over x squared in this form. Well, this one is pretty straightforward because we're dividing by x squared. So you could literally view this as 7x to the sixth divided by x squared plus x to the third divided by x squared plus 2x divided by x squared plus 1 divided by x squared. So we could just do this term by term. What's 7x to the sixth divided by x squared? Well, x to the sixth divided by x squared is x to the fourth. So it's going to be 7x to the fourth power. And then, same thing right over here. Plus x to the third divided by x squared. Well, that's just going to be x. So plus x. And then, we're going to have 2x divided by x squared. But remember, we want to write it in a form of r of x over b of x-- where r of x has a lower degree than b of x. Well, 2x has a lower degree than x squared. Here this is degree 1. This is degree 2. So you could write it as plus 2x over x squared. Like that. And then, you could write plus 1 over x squared. So you could do this-- plus 1 over x squared. So you could write it like that. But that's not exactly the form that they want. They want us to write it q of x-- and you could view that as 7x to the fourth plus x. And then, they want plus r of x over b of x So plus some polynomial over x squared in this case. So instead of writing it as 2x over x squared plus 1 over x squared, we could just write it as 2x plus 1 over x squared. So one way to think about it. So let me just put some parentheses here so that it interprets my typing correctly. So notice, this part of the polynomial, these terms have an equal or higher degree than x squared. So I just divided those. 7x to the sixth divided by x squared is 7x to the fourth. x to the third divided by x squared is x. And then, once I got two terms that had a lower degree than x squared, I just left on there. I just said plus whatever 2x plus 1 divided by x squared is. And that's the form that they wanted us to write it in. We'll check our answer. And we got it right.