When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder?  Enter all the possible values, separated by commas.
In general, when a polynomial is divided by a polynomial of degree $d,$ then the possible degrees of the remainder are 0, 1, 2, $\dots,$ $d - 1.$  Therefore, the possible degrees of the remainder here are $\boxed{0,1}.$
01
Level 2
Intermediate Algebra
Problem: When a polynomial is divided by $2x^2 - 7x + 18,$ what are the possible degrees of the remainder?  Enter all the possible values, separated by commas.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the possible degrees of the remainder when dividing a polynomial by another polynomial, we apply the polynomial remainder theorem. This theorem states that when a polynomial \( P(x) \) is divided by a polynomial \( D(x) \), the remainder \( R(x) \) will have a degree that is less than the degree of \( D(x) \).

First, we identify the degree of the divisor polynomial \( 2x^2 - 7x + 18 \). The highest power of \( x \) in this polynomial is \( x^2 \), which indicates that the degree of \( D(x) \) is 2.

According to the polynomial remainder theorem, the degree of the remainder \( R(x) \) must be less than the degree of the divisor. Therefore, the possible degrees for the remainder can be:

1. The degree can be 0, which corresponds to a constant remainder.
2. The degree can be 1, which corresponds to a linear remainder.

Since the degree of the remainder must be less than 2, the only possible degrees for the remainder are 0 and 1.

Thus, the final answer, representing the possible degrees of the remainder, is 0, 1.<|END_OF_LATENT|> The final answer is $01$.