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 Q of X and RN 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 6 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 6 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 6 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 2 of a 2x has a lower degree than x squared here this is a 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 is to write it Q of X and that could be you could view that as 7x to the fourth plus X and then they want to 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 Pala know these these terms have an equal or higher degree than x squared so I just divided those 7x is 6 divided by x squared is 7x to the fourth X to the 3rd divided by x squared is X and then once I got two terms that had a lower degree the next squared I just left them 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