Enter all but one of the polynomial function, root, quotient polynomial, and remainder into the Factor Theorem Calculator formula to determine if a given value is a root of the polynomial function.

## Factor Theorem Formula

f(x) = (x - a) cdot q(x) + r

## What is a Factor Theorem?

The Factor Theorem is a mathematical theorem that provides a criterion for determining whether a given binomial is a factor of a given polynomial. It states that a polynomial f(x) has a factor (x – c) if and only if f(c) = 0. In other words, if a certain value, when substituted into the polynomial, results in zero, then the corresponding binomial is a factor of that polynomial. This theorem is a special case of the polynomial remainder theorem and is used in polynomial division and in finding the roots of a polynomial equation. It is a fundamental tool in algebra and calculus, and it simplifies the process of factoring polynomials and solving polynomial equations.

## How to Calculate Factor Theorem?

The following steps outline how to use the Factor Theorem:

1. First, determine the polynomial function.
2. Next, find a potential factor of the polynomial function.
3. Next, use synthetic division to divide the polynomial function by the potential factor.
4. Finally, analyze the result to determine if the potential factor is a factor of the polynomial function.
5. If the potential factor is a factor, repeat the process with the quotient obtained from the synthetic division.
6. If the potential factor is not a factor, try another potential factor.
7. Continue this process until all potential factors have been tested.
8. Once all potential factors have been tested, the remaining polynomial function is the quotient obtained from the synthetic division.

Example Problem:

Use the following polynomial function as an example problem to test your knowledge:

Polynomial function: f(x) = 2x^3 – 5x^2 + 3x – 1

Potential factor: x – 1