Remainder Theorem Calculator
Apply the Remainder Theorem to find remainders and perform polynomial division with comprehensive analysis. Our algebra calculator supports polynomial evaluation, factorization checking, synthetic division, and complete remainder analysis.
Last updated: December 15, 2024
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Enter polynomial with standard notation: x^3, x^2, x, constants
Enter as (x - a), (x + a), or just the value a
Division:
(x^3 - 2x^2 + x - 1) ÷ (x - 2)
Remainder Analysis
Remainder:
1
Analysis Type:
Polynomial evaluation
Quotient:
Q(x) = x^2 + 1
Verification:
P(2) = 1
Factorization:
x^3 - 2x^2 + x - 1 = (x - 2)(x^2 + 1) + 1
Solution Steps:
- 1. Apply Remainder Theorem: P(a) gives the remainder when P(x) is divided by (x - a)
- 2. For divisor (x - 2), evaluate P(2)
- 3. P(2) = 2^3 - 2(2^2) + 2 - 1
- 4. P(2) = 8 - 8 + 2 - 1 = 1
Analysis:
Using the Remainder Theorem, P(2) = 1. This means when the polynomial is divided by (x - 2), the remainder is 1.
Remainder Theorem Tips:
- • Theorem: When P(x) is divided by (x - a), remainder = P(a)
- • Factor Test: If P(a) = 0, then (x - a) is a factor of P(x)
- • Synthetic Division: Efficient method for linear divisors
- • Applications: Finding roots, factoring, and polynomial evaluation
Quick Example Result
For (x³ - 2x² + x - 1) ÷ (x - 2):
Remainder = 1
P(2) = 8 - 8 + 2 - 1 = 1
How This Calculator Works
Our Remainder Theorem calculator applies the fundamental principle that when a polynomial P(x) is divided by (x - a), the remainder equals P(a). The calculator performs polynomial evaluation, synthetic division, and factorization analysisto provide comprehensive insights into polynomial division and factor relationships.
Remainder Theorem Algorithm
Extract coefficients and identify degreeFrom (x - a), identify the value aCalculate P(a) to find remainderFind quotient using efficient algorithmThe Remainder Theorem algorithm systematically evaluates polynomials at specific points to determine division remainders. For a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ divided by (x - c), the remainder is P(c) = aₙcⁿ + aₙ₋₁cⁿ⁻¹ + ... + a₁c + a₀. Synthetic division provides an efficient method to find both quotient and remainder simultaneously.
Division algorithm and factor theorem relationship
Mathematical Foundation
The Remainder Theorem is a fundamental result in algebra that connects polynomial division with polynomial evaluation. It states that the remainder when P(x) is divided by (x - a) is exactly P(a). This theorem is closely related to the Factor Theorem, which states that (x - a) is a factor of P(x) if and only if P(a) = 0. These theorems form the basis for polynomial factorization, root finding, and many applications in algebra and calculus.
- Polynomial division algorithm provides the theoretical foundation
- Synthetic division offers computational efficiency for linear divisors
- Factor theorem enables systematic root finding and factorization
- Applications extend to interpolation, approximation, and numerical analysis
Sources & References
- Abstract Algebra - David S. Dummit and Richard M. FooteComprehensive treatment of polynomial rings and division algorithms
- Wolfram MathWorld - Remainder Theorem ReferenceDetailed mathematical reference with examples and applications
- College Algebra - Robert F. BlitzerEducational treatment of polynomial division and remainder theorem applications
Need help with other polynomial operations? Check out our polynomial calculator and factoring calculator.
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Problem Setup:
- Polynomial: P(x) = 2x³ - 5x² - x + 6
- Test Factor: (x - 3)
- Method: Remainder Theorem
- Question: Is (x - 3) a factor?
Solution Process:
Result: P(3) = 12 ≠ 0, so (x - 3) is NOT a factor
Since P(3) = 12 ≠ 0, the Remainder Theorem tells us that (x - 3) is not a factor of the polynomial 2x³ - 5x² - x + 6. The remainder when dividing by (x - 3) is 12. This demonstrates how the Remainder Theorem provides a quick test for polynomial factors. If we had found P(3) = 0, then (x - 3) would be a factor, and we could use synthetic division to find the quotient polynomial. The complete division gives us: 2x³ - 5x² - x + 6 = (x - 3)(2x² + x + 2) + 12, confirming our remainder calculation.
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