When solving problems regarding polynomial remainder theorem, there are things that we need take into account the characteristic of remainder with respect to the divisor.
Let P(x) be the Polynomial with degree n and D(x) be the divisor with degree r such that n and r is the elements of all positive integer and n>r>1. If P(x) is divided by D(x), the remainder R(x) has a degree of r-1.
If the divisor is x-c, this is a binomial with degree 1. Then, the remainder must have a degree of 1-1=0. This can be written as Ax0 or A, a constant.
If the divisor is in the form of , a quadratic in degree 2. The remainder must have a a degree of 2-1=1. A binomial in the form of Ax+c.
If the divisor has a degree of 4, the remainder is cubic.
Why do we need to know this?
This is for a problem solver or a student to increase their accuracy. There are questions in Polynomial remainder theorem that this technique is really helpful to quickly at the solution. Most of its application can be seen in Remainder theorem with 2nd degree divisor in this page.