When $x^3+x^2-7$ is divided by $(x-3)$, my book states that because the polynomial is cubic and the divisor is linear then the quotient must be quadratic and the remainder must be a constant. Basically, it sets put the following: $x^3+x^2-7=(Ax^2+Bx+C)\cdot(x-3)+D$
In another example, the book divides $x^4+x^3+x-10$ by $x^2+2x-3$, and states that because the divisor is quadratic and because the polynomial is quartic then quotient must be a quadratic and the remainder must be a linear expression.
I want to understand what logic they use here, in order to decide the morphology of the equation whose coefficients they will begin to find. I have been going at this question for a long time. No online teacher seems to tackle it, although it seems like a bog issue.
Another reason why I really want to understand this is because it arises in partial fractions, such as the partial fraction involved in the first order differential equation in question 7 of this booklet (https://madasmaths.com/archive/maths_booklets/further_topics/integration/1st_order_differential_equations_substitutions.pdf). The logic is even more baffling. Someone please help, I have a further maths exam coming up in 3 days and this isn't difficult, but just obscure.
[Note: please note that I don't want some complex mathemtical proof. I just want know the logic so that I can, with reasonable simplicity, be able to explain to someone else how to decide upon the morphology of the the right hand side when using the factor theorem and factorising a polynomial by inspection, in order to find the coefficients.]