Does Division of Polynomials Follow a Pattern of Degree Decrease?

[chwala]

When you have a polynomial say ax^4+bx^3+cx^2+dx+e where a,b,c,d and e are constants and divide this by a polynomial say ax+b it follows that the quotient will be a cubic polynomial. Assuming that a remainder exists, then the remainder will be a constant because in my reasoning, the remainder should be 1 degree less than the divisor ax+b.
My question is supposing you have a polynomial of degree 7 and divide this by a cubic divisor of degree 3 then it follows that the quotient will be quartic of degree 4, does it follow that the remainder if it exists will be quadratic of degree 2? I need insight on this.

 

[symbolipoint]

If a remainder occurs, it has denominator equal to the divisor!

 

[SteamKing]

Your insight is here:

http://www.sosmath.com/algebra/factor/fac01/fac01.html

 

[chwala]

Thanks i have looked at the insight and i do appreciate. My interest is on the relation between the divisor and the remainder, i am trying to come up with a general way of expressing this. Supposing you have say a polynomial of degree 35 and divide this by say a polynomial of degree 27, my understanding is that the quotient will be of degree 8. My concern is on the remainder, supposing a remainder exists will it be of degree 26.

 

[chwala]

Thanks, my interest is on the relationship between the divisor and the remainder, supposing you have a polynomial, say of degree 66 and a polynomial divisor of say degree 41, in my understanding the quotient polynomial will be of degree 25. Does it also follow that the Remainder will be of degree 40,( one degree less than divisor)?

 

[symbolipoint]

The remainder will have a degree of the numerator less than the degree of the denominator. The denominator is the same as the divisor.