Factor Theorem Calculator
Enter a polynomial's coefficients and a candidate root c to test whether (x − c) is a factor, with synthetic division table and quotient polynomial.
Verdict
Synthetic Division Table
Step-by-Step
What Is the Factor Theorem?
The Factor Theorem connects polynomial roots and factors: a polynomial f(x) has (x − c) as a factor if and only if plugging in c gives zero. This simple yet powerful idea lets you test potential roots systematically — evaluate f(c), and if the result is zero, you know immediately that (x − c) divides f(x) without any remainder.
In practice, you combine the Factor Theorem with synthetic division. Once you confirm that f(c) = 0, synthetic division quickly produces the quotient polynomial. For a cubic, this reduces the problem to factoring a quadratic. For higher-degree polynomials, you repeat the process with new candidate roots until the polynomial is fully factored.
Factor Theorem vs. Remainder Theorem
Remainder Theorem
f(x) ÷ (x−c) gives remainder f(c)
Factor Theorem
If f(c) = 0, then (x−c) is a factor
How to Use the Factor Theorem Calculator
Test any root candidate and perform synthetic division in three steps.
Enter Coefficients & Root
Type the polynomial coefficients from highest to lowest degree, separated by commas, and the candidate root c.
Click "Test Factor"
The engine evaluates f(c) via synthetic division and determines whether (x − c) is a factor.
Read the Result
See the verdict, synthetic division table, quotient polynomial, and remainder.
Worked Examples
Four Factor Theorem problems with synthetic division.
1Is (x−1) a factor of x³−6x²+11x−6?+−
2Is (x−4) a factor of x³−6x²+11x−6?+−
3Is (x+2) a factor of x³+2x²−5x−6?+−
4Is (x−3) a factor of 2x³−7x²+2x+3?+−
Key Theorems Reference
Theorems related to polynomial roots and factors.
| Theorem | Statement | Use |
|---|---|---|
| Factor Theorem | (x−c) | f(x) ⟺ f(c) = 0 | Test if c is a root |
| Remainder Theorem | f(x) ÷ (x−c) → remainder = f(c) | Find remainder fast |
| Rational Root | Candidates: ±(factors of aₙ)/(factors of a₀) | List possible roots |
| Fundamental Theorem of Algebra | Degree n → exactly n roots (counting multiplicity) | Count expected roots |
Frequently Asked Questions
Common questions about the Factor Theorem.
What is the Factor Theorem?+−
The Factor Theorem states that (x − c) is a factor of polynomial f(x) if and only if f(c) = 0. In other words, c is a root of the polynomial exactly when (x − c) divides it evenly.
How does synthetic division work?+−
Synthetic division is a streamlined version of long division for dividing a polynomial by (x − c). You bring down the leading coefficient, multiply by c, add to the next coefficient, and repeat until you reach the remainder.
What is the Rational Root Theorem?+−
It states that any rational root p/q of a polynomial with integer coefficients must have p dividing the constant term and q dividing the leading coefficient, giving a finite list of candidates to test.
Can the Factor Theorem work for non-integer roots?+−
Yes — the theorem applies to any value c (integer, rational, irrational, or complex). However, this calculator focuses on integer and simple rational candidates.
What happens when f(c) ≠ 0?+−
If f(c) ≠ 0, then (x − c) is NOT a factor of f(x). The value f(c) is the remainder when dividing f(x) by (x − c), by the Remainder Theorem.
How is the Factor Theorem related to the Remainder Theorem?+−
The Factor Theorem is a special case of the Remainder Theorem. The Remainder Theorem says dividing f(x) by (x − c) gives remainder f(c). When that remainder is 0, (x − c) is a factor.
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