The remainder in polynomial long division is the polynomial left over after the division process that has a degree less than the divisor. It represents what's "left over" that can't be evenly divided by the divisor.
Here's a more detailed explanation:
When you perform polynomial long division, you're trying to divide one polynomial (the dividend) by another (the divisor) to find a quotient and a remainder, similar to how you perform long division with numbers.
The general relationship is:
Dividend = (Divisor * Quotient) + Remainder
Therefore:
- Quotient: The result of the division (the polynomial above the division line).
- Remainder: The polynomial left over after the division process, which has a lower degree than the divisor. It represents the portion of the dividend that could not be evenly divided by the divisor. We often express the answer as:
quotient + remainder/divisor.
Analogy to Numerical Long Division:
Think about dividing 17 by 5 using long division.
- 5 goes into 17 three times (3 is the quotient).
- 3 * 5 = 15
- 17 - 15 = 2 (2 is the remainder).
This means 17 = (5 * 3) + 2. We can express the answer as 3 + 2/5.
Example with Polynomials:
Let's say we divide x^2 + 3x + 5 by x + 1. After performing the long division, we might find:
- Quotient:
x + 2 - Remainder:
3
This means x^2 + 3x + 5 = (x + 1)(x + 2) + 3. We can express the answer as (x + 2) + 3/(x + 1).
Significance of the Remainder:
- Factor Theorem: If the remainder is 0, it means the divisor is a factor of the dividend.
- Evaluating Polynomials: The Remainder Theorem states that if you divide a polynomial f(x) by x - a, the remainder is f(a).
- Simplifying Rational Expressions: Long division can help simplify rational expressions (polynomials divided by polynomials) by expressing them as the sum of a polynomial and a simpler rational expression.
