Remainder theorem Calculator
Find the remainder when a polynomial is divided by (x−a).
Formula first
Overview
The Remainder Theorem provides a direct method for calculating the remainder of a polynomial division by a linear factor without performing the full division process. It states that when a polynomial f(x) is divided by a linear divisor in the form (x - a), the remainder is equal to the value of the polynomial evaluated at a.
Symbols
Variables
a = Value a, f(a) = Function Value
Apply it well
When To Use
When to use: Use this theorem when you need to find the remainder of a polynomial division specifically by a linear divisor of the form (x - a). It is a highly efficient shortcut when the quotient itself is not required, saving time compared to long or synthetic division.
Why it matters: This theorem forms the logical basis for the Factor Theorem, which is essential for solving algebraic equations and factoring high-degree polynomials. In computational science, it helps in the development of algorithms for error detection and polynomial interpolation.
Avoid these traps
Common Mistakes
- Evaluating f(-a) instead of f(a).
- Confusing divisor (x-a).
One free problem
Practice Problem
Calculate the remainder when the polynomial f(x) = x³ - 4x² + 2x - 5 is divided by (x - 3).
Solve for:
Hint: Substitute the value x = 3 into the function f(x).
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Remainder theorem
- Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
- Britannica: Remainder theorem
- Stewart, James. Calculus: Early Transcendentals
- Stewart, Redlin, and Watson, Precalculus: Mathematics for Calculus, 7th ed.
- OCR A-Level Mathematics — Pure (Algebra)