The Factor Theorem Calculator is a digital math tool designed to test whether a linear binomial (x − c) is a factor of a given polynomial. It applies the factor theorem, which is a key concept in algebra used to simplify expressions and solve polynomial equations. With this calculator, students, educators, and professionals can quickly determine if substituting a value into the polynomial results in zero—indicating a valid factor. This tool is useful in polynomial division, root-finding, and algebraic verification tasks.
This calculator belongs to the algebraic polynomial and root-solving tools category. It’s especially helpful in academic environments and in mathematical software development for symbolic computation.
formula of Factor Theorem Calculator
Factor Theorem Statement:
If f(c) = 0, then (x − c) is a factor of the polynomial f(x)
Step-by-step Process
Let f(x) be a polynomial of degree n:
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
- Choose a value
c - Substitute
cinto the polynomial:
f(c) = aₙ(c)ⁿ + aₙ₋₁(c)ⁿ⁻¹ + … + a₁(c) + a₀ - Check the result:
- If f(c) = 0 → (x − c) is a factor
- If f(c) ≠ 0 → (x − c) is not a factor
This method avoids long polynomial division and helps identify possible roots or simplifications quickly.
Common Factor Theorem Lookup Table
| Polynomial Expression | Test Value (c) | f(c) Result | Is (x − c) a Factor? |
|---|---|---|---|
| f(x) = x² − 5x + 6 | 2 | 0 | Yes |
| f(x) = x³ + x² − x − 1 | 1 | 0 | Yes |
| f(x) = 2x² + 3x + 4 | −2 | 6 | No |
| f(x) = x³ − 7x + 6 | 3 | 0 | Yes |
This table helps users identify common outcomes for standard polynomials without redoing the work each time.
Example of Factor Theorem Calculator
Let’s check if (x − 2) is a factor of f(x) = x³ − 4x² + x + 6.
Step 1: Define the polynomial
f(x) = x³ − 4x² + x + 6
Step 2: Substitute c = 2 into the polynomial
f(2) = (2)³ − 4(2)² + (2) + 6
f(2) = 8 − 16 + 2 + 6 = 0
Step 3: Since f(2) = 0, we conclude:
(x − 2) is a factor of f(x).
This means x = 2 is a root of the polynomial.
Most Common FAQs
The Factor Theorem tells us when (x − c) is a factor by checking if f(c) = 0. The Remainder Theorem tells us what the remainder is when dividing by (x − c), which is simply f(c).
Yes, the Factor Theorem works for any real or complex number. If f(c) = 0 for that number, then (x − c) is a valid factor.
It simplifies polynomial factorization, making it easier to solve equations, analyze graphs, or perform synthetic division without manually dividing.