Remainder and Factor Theorem Calculator

Enter coefficients or an expression to analyze divisibility and remainders with precision. Check candidate roots, compute P(a), and confirm factors using thresholds for robustness. See synthetic division steps, quotient polynomials, and evaluation logs instantly for transparency. Export tables as CSV, print reports to PDF beautifully anywhere. Master remainders and factors with guided clarity now.

Input
Switch between expression or direct coefficients.
Values |P(a)| ≤ tolerance count as zero.
Use terms like 3x^4 - 2x + 7. Implicit 1 allowed: -x^3 + x - 5.
Results
P(x) = 2*x^3 - 6*x^2 + 2*x - 1 Degree: 3
# a P(a) Is factor? Quotient polynomial Steps
1 1 -3 No 2*x^2 - 4*x - 2
Synthetic division steps for a = 1
Index Bring down Multiply by a Add coefficient Running value
0 2 2
1 2 -6 -4
2 -4 2 -2
3 -2 -1 -3
2 2 -5 No 2*x^2 - 2*x - 2
Synthetic division steps for a = 2
Index Bring down Multiply by a Add coefficient Running value
0 2 2
1 4 -6 -2
2 -4 2 -2
3 -4 -1 -5
3 3 5 No 2*x^2 + 2*x^-1
Synthetic division steps for a = 3
Index Bring down Multiply by a Add coefficient Running value
0 2 2
1 6 -6 0
2 0 2 2
3 6 -1 5
No exact linear factors detected under the current tolerance.
Formula Used

Remainder Theorem: Dividing P(x) by (x − a) leaves remainder P(a).

Factor Theorem: (x − a) is a factor of P(x) exactly when P(a) = 0.

Horner’s Method: Efficient evaluation using recurrence b0=an, bk=a·bk−1+an−k.

We detect factors by |P(a)| ≤ tolerance.

How to Use
  1. Choose input mode and enter P(x) or coefficients.
  2. Enter candidate a values, or enable auto rational testing.
  3. Set a sensible zero tolerance for numeric stability.
  4. Click Evaluate to compute P(a) and factors.
  5. Expand Steps to see synthetic division details.
  6. Export the results table as CSV or print to PDF.

For non-integer coefficients, auto rational candidates may miss true roots.

Example Data Table
P(x)aP(a)Is factor?
x^3 - 6x + 820Yes
x^3 - 6x + 813No
x^3 - 6x + 8-2-4No
This example shows the remainder test for common a values.
FAQs

It is the remainder when dividing P(x) by (x − a).

Use small values like 1e-9 for numeric robustness.

Rational testing targets ratios of integers by design.

This tool checks linear factors; use factoring tools otherwise.

Division by (x − a) lowers degree by one when remainder is zero.

Yes, enter coefficients in descending powers of x.

This tool focuses on real a; complex factors need other methods.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.