Factor Theorem Calculator
Checking factors of a polynomial.
Formula first
Overview
The Factor Theorem is a specialized application of the Remainder Theorem which establishes a direct link between the roots of a polynomial and its linear factors. It states that a polynomial f(x) has (x - a) as a factor if and only if the function evaluated at 'a' results in a remainder R equal to zero.
Symbols
Variables
R = Remainder, f(a) = Value of f(a)
Apply it well
When To Use
When to use: Use this theorem when you need to verify if a binomial is a factor of a polynomial without performing long division. It is the primary method for finding roots of higher-degree polynomials and simplifying complex algebraic expressions.
Why it matters: This theorem is a fundamental tool in algebra and calculus, enabling the decomposition of complex functions into simpler linear components. It is essential for solving polynomial equations and understanding the behavior of functions at their x-intercepts.
Avoid these traps
Common Mistakes
- Sign errors: factor is (x-a), root is a.
- Not checking remainder is exactly 0.
One free problem
Practice Problem
A student uses the Factor Theorem to check if (x - 3) is a factor of f(x) = x² - 9. If the remainder R is calculated to be 0, what is the value of the function evaluation fa?
Solve for: R
Hint: The theorem states that the function evaluation f(a) is identical to the remainder R.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Britannica: Factor Theorem
- Wikipedia: Factor Theorem
- Wikipedia: Remainder Theorem
- Britannica article: Factor theorem
- Standard curriculum — A-Level Pure Mathematics (Algebra)