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:

- First, determine the polynomial function.
- Next, find a potential factor of the polynomial function.
- Next, use synthetic division to divide the polynomial function by the potential factor.
- Finally, analyze the result to determine if the potential factor is a factor of the polynomial function.
- If the potential factor is a factor, repeat the process with the quotient obtained from the synthetic division.
- If the potential factor is not a factor, try another potential factor.
- Continue this process until all potential factors have been tested.
- 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