Question

what can be the degree of the remainder at most when a fourth degree polynomial is divided by a quadratic polynomial

Answer 1

degree 2 because 4 divided by 2 is 2

Answer 2

Remainder is of 1 degree when fourth degree polynomial is divided by quadratic polynomial.

Solution:

Let us take a random 4-degree equation. Like $x^{4}+x^{3}+x^{2}+1$ and a quadratic equation as $x^{2}+x+1$

We get the solution be dividing $\frac{x^{4}+x^{3}+x^{2}+1}{x^{2}+x+1}$ .The division process is shown in below attached image.

According to the above division we can see that the remainder is 1. Hence, degree can be as small as 1. Another fact is the degree of remainder is always less than that of divisor since the divisor has only 2 degree.

Therefore the remainder can only be 1 degree.