Polynomial Division
What is the Polynomial Division Algorithm?
Polynomial division is similar to regular long division but with polynomials instead of numbers. It allows us to divide one polynomial (dividend) by another polynomial (divisor) to get a quotient and remainder. The algorithm states that for polynomials A(x) and D(x), there exist unique polynomials Q(x) and R(x) such that: A(x) = D(x)Q(x) + R(x), where degree of R(x) < degree of D(x).
Key Components:
A(x) = D(x)Q(x) + R(x)
1. Dividend A(x): The polynomial being divided
2. Divisor D(x): The polynomial we divide by
3. Quotient Q(x): Result of division
4. Remainder R(x): What's left over
Key Rules:
• Degree of remainder must be less than divisor
• Terms must be arranged in descending order
• Process continues until no further division possible
Example:
Divide A(x) = x³ + 2x² - x - 2 by D(x) = x + 1
Process:
1. Divide first terms: x³ ÷ x = x²
2. Multiply: x²(x + 1) = x³ + x²
3. Subtract: (x³ + 2x² - x - 2) - (x³ + x²) = x² - x - 2
4. Repeat with x²
5. Continue until remainder degree < divisor degree
Final Result:
x³ + 2x² - x - 2 = (x + 1)(x² + x - 2) + 0
where:
• Q(x) = x² + x - 2
• R(x) = 0