Which of the following describes the remainder in polynomial long division?

When performing polynomial long division, the remainder indeed has a specific relationship with the divisor. After dividing the polynomials, any remainder can be expressed as a fraction where the numerator is the remainder itself and the denominator is the divisor. This allows us to write the division result in a more complete mathematical form, typically as:

[

\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}

]

This representation is essential as it shows how the remainder affects the overall outcome of the division. In situations where the remainder is non-zero, this fractional representation indicates that the original polynomial cannot be evenly divided by the divisor, providing a clearer understanding of the results.

The other options do not accurately reflect the nature of the remainder in polynomial long division. For example, leaving the remainder out will not accurately represent the results, and a remainder is not always zero unless the divisor perfectly divides the polynomial. Additionally, the idea of multiplying the remainder by the dividend does not apply, as the process of division focuses on how many times the divisor fits into the dividend rather than modifying the dividend itself by the remainder.