How do you write a remainder in algebraic division?

When performing algebraic division, the remainder is written as a fraction over the divisor. Here's a detailed explanation using the formula:

Understanding Algebraic Division and Remainders

In algebraic division, we divide a polynomial (the dividend) by another polynomial (the divisor). The result of this division gives us a quotient and potentially a remainder. The fundamental relationship is:

Dividend = (Divisor × Quotient) + Remainder

This formula helps verify the accuracy of the division.

How to Express the Remainder

The remainder in algebraic division is typically expressed in one of two ways:

Separate Remainder: This is written as a separate term, often after the quotient.
Fractional Remainder: This is written as a fraction where the remainder becomes the numerator and the divisor becomes the denominator. This fractional remainder is then added to the quotient.

Method 1: Separate Remainder

After performing the polynomial long division, you end up with a quotient and a remainder.
The final result is expressed as "Quotient + Remainder".
- For example, if dividing x² + 3x + 5 by x+1 results in a quotient of x + 2 and a remainder of 3, the expression would be written as x + 2 + 3.

Method 2: Fractional Remainder

The remainder becomes the numerator of a fraction.
The divisor becomes the denominator of that fraction.
The final result is expressed as "Quotient + (Remainder/Divisor)".
- For example, using the same quotient and remainder as above, x² + 3x + 5 divided by x+1 would be expressed as x + 2 + 3/(x+1).
- This is the more common way to represent remainders in polynomial division.

Examples

Let’s consider an example using the reference's remainder theorem in practice.

Suppose we divide p(x) = x² + 3x + 5 by the linear polynomial x + 1. We use the zero of the linear polynomial which is x = -1, so k=-1.

p(-1) = (-1)² + 3(-1) + 5 = 1 - 3 + 5 = 3

This is the remainder, as specified by the remainder theorem when the divisor is linear.

Performing the long division would give:

*   **Quotient:** x + 2
*   **Remainder:** 3

The complete expression using fractional remainder is: **x + 2 + 3/(x+1)**

Table Summary

Expression Type	How the Remainder is Written	Example from above
Separate	Quotient + Remainder	x + 2 + 3
Fractional	Quotient + (Remainder / Divisor)	x + 2 + 3/(x+1)

Key Points

The remainder theorem, mentioned in the reference, provides an easy method to find the remainder when dividing a polynomial by a linear factor x-k, where the remainder is p(k). This works only with linear divisors.
If the divisor is not linear, you need to use long division or synthetic division.
The fractional remainder expression is more often used in algebraic contexts to represent the complete result.

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