How do you express the remainder in polynomial long division?

In polynomial long division, expressing the remainder correctly is essential for maintaining the integrity of the division process. When you divide one polynomial by another, after performing the long division, you will arrive at a quotient and a remainder.

The proper way to represent this remainder is to express it as a fraction that is placed over the original divisor. This means that the final expression of the division takes the form of the quotient plus the remainder divided by the divisor. This format clearly illustrates the entire result of the division, ensuring that all values are accurately represented.

For instance, if you divide ( P(x) ) by ( D(x) ) and obtain a quotient ( Q(x) ) and a remainder ( R(x) ), the complete expression would be ( Q(x) + \frac{R(x)}{D(x)} ). This method of expressing the remainder maintains the structural relationship of the polynomials and facilitates further calculations or evaluations based on this division.

Other approaches, such as placing the remainder simply as a whole number or ignoring it altogether, do not provide a comprehensive understanding of the division's outcome. Similarly, adding the remainder directly to the quotient without the divisor does not maintain mathematical accuracy in the expression of the result. Hence,