Question

question: if a polynomial of degree 8 is divided by a polynomial of degree 5, state the largest possible degree of the remainder. 1) if a polynomial of degree 8 is divided by a polynomial of degree 5, state the largest possible degree of the remainder. options: 5, 1, none of these options, 8, 4, 6, 2, 3, all of the other options (except none of these options), skip this question, not enough information, 7

Explanation:

Step1: Recall Polynomial Division Rule

When a polynomial \( f(x) \) of degree \( n \) is divided by a polynomial \( g(x) \) of degree \( m \) (\( m \leq n \)), the degree of the quotient \( q(x) \) is \( n - m \), and the degree of the remainder \( r(x) \) satisfies \( \text{deg}(r(x)) < \text{deg}(g(x)) \).

Step2: Apply the Rule to the Problem

Here, \( \text{deg}(f(x)) = 8 \) and \( \text{deg}(g(x)) = 5 \). So, the degree of the remainder \( r(x) \) must be less than 5. The largest integer less than 5 is 4? Wait, no, wait. Wait, the divisor is degree 5, so the remainder's degree must be less than 5, so the maximum possible degree of the remainder is \( 5 - 1 = 4 \)? Wait, no, let's recheck.

Wait, the formula is: if \( f(x) = g(x)q(x) + r(x) \), then \( \text{deg}(r(x)) < \text{deg}(g(x)) \). So \( \text{deg}(g(x)) = 5 \), so \( \text{deg}(r(x)) < 5 \). The largest integer less than 5 is 4? But wait, the options include 3, 2, 6, 4, 8, 1, 5, 7, etc. Wait, no, maybe I made a mistake. Wait, no, the divisor is degree 5, so the remainder's degree must be less than 5, so the maximum possible degree is 4. Wait, but the options have 4 as an option (the circle labeled 4). Wait, let's confirm again.

Suppose we have \( f(x) \) degree 8, \( g(x) \) degree 5. Then \( q(x) \) is degree \( 8 - 5 = 3 \), and \( r(x) \) is degree less than 5. So the maximum degree of \( r(x) \) is 4. So the largest possible degree of the remainder is 4. So the answer should be 4.

Answer:

4 (the option with the circle labeled 4)