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Factor Theorem Calculator

Check whether x - c is a factor of P(x)

Polynomial coefficients

Test factor x - c, c =

Factor Theorem Formula

x-c is a factor of P(x) ⇔ P(c)=0

The factor theorem is the zero-remainder case of the remainder theorem. To test whether x-c divides a polynomial evenly, substitute c into the polynomial and compute P(c). If the value is zero, the division has no remainder and x-c is a factor. If the value is not zero, the same value is the remainder and the candidate expression is not a factor.

⚠Note: This calculator tests linear factors of the form x-c.

How the Test Works

The factor theorem is a shortcut for checking linear factors. If substituting c into the polynomial gives zero, then x-c divides the polynomial with no remainder.

Evaluate P(c)

Substitute c into the polynomial.

Zero Remainder

If P(c)=0, the factor divides evenly.

Root Connection

P(c)=0 also means c is a root or zero of the polynomial.

Remainder Check

If P(c) is not zero, that value is the remainder.

💡 Example: For P(x)=x³-3x²+2x and c=1, P(1)=0, so x-1 is a factor.

Applications

FactoringRoot TestingSynthetic Division

Frequently Asked Questions

What is a factor theorem calculator?▼

It checks whether x-c is a factor of a polynomial by calculating P(c).

What is the factor theorem?▼

The factor theorem says x-c is a factor of P(x) if and only if P(c)=0.

How do I test x+4?▼

Since x+4 equals x-(-4), enter c=-4.

What if P(c) is not zero?▼

If P(c) is not zero, x-c is not a factor and P(c) is the remainder.

How is this different from the remainder theorem?▼

The factor theorem is the zero-remainder case of the remainder theorem.

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