Polynomial Factoring Calculator

Calculator Inputs

Polynomial degree

Variable symbol

Graph minimum x

Coefficient a

Use integers, decimals, or fractions.

Coefficient b

Coefficient c

Coefficient d

Coefficient e

Graph maximum x

Formula used

This page works with the standard polynomial form:

P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

It then applies these algebra ideas:

1) Pull out the greatest common factor when possible.
2) Test rational roots using p/q from the Rational Root Theorem.
3) Use synthetic division after each confirmed root.
4) Continue until only a linear or irreducible remainder stays.

The Rational Root Theorem uses:

Possible rational roots = ± (factors of constant term) / (factors of leading coefficient)

The calculator factors over rational numbers when possible. If a part does not factor nicely over rationals, it stays as the remaining polynomial.

How to use this calculator

Select degree 2, 3, or 4.
Enter the coefficients in descending order of powers.
Choose the variable symbol and graph interval.
Press Factor Polynomial.
Review the factored form, rational roots, remaining part, and graph.
Use the CSV or PDF buttons to export your result summary.

Example data table

Polynomial	Degree	Expected factorization	Roots	Note
x² - 5x + 6	2	(x - 2)(x - 3)	2, 3	Classic quadratic with two rational roots.
x³ - 6x² + 11x - 6	3	(x - 1)(x - 2)(x - 3)	1, 2, 3	Good test case for repeated synthetic division.
2x² - 5x + 2	2	2(x - 1/2)(x - 2)	1/2, 2	Shows a rational root with a fractional value.
x⁴ - 5x² + 4	4	(x - 2)(x - 1)(x + 1)(x + 2)	-2, -1, 1, 2	Quartic example with four real rational roots.

FAQs

1) What kinds of polynomials can this calculator factor?

It handles quadratic, cubic, and quartic expressions entered by coefficients. It looks for rational factors, extracts common factors, and leaves any irreducible remainder in polynomial form when needed.

2) Does it always fully factor every polynomial?

No. It fully factors many expressions over rational numbers. Some polynomials contain irrational or complex factors, so the calculator may leave a remaining unfactored part instead of forcing an incorrect factorization.

3) Can I enter fractions or decimals as coefficients?

Yes. You can enter values like 3, -4.5, or 7/3. The calculator converts them into exact rational values for the factoring checks whenever possible.

4) Why does the result sometimes show a scale factor first?

The tool extracts a common numeric multiplier before factoring the rest. That keeps the inner polynomial simpler and makes the factorization easier to read and verify.

5) What does the candidate roots row mean?

That row lists the rational values suggested by the Rational Root Theorem. The calculator tests those values and confirms which ones actually make the polynomial equal zero.

6) Why are approximate real roots shown too?

Approximate roots help you connect the algebra with the graph. They are useful when a remaining factor does not split nicely into rational linear factors.

7) How should I enter coefficients correctly?

Enter them in descending power order. For a cubic, use coefficients for x³, x², x, and the constant term. For a quadratic, use x², x, and constant.

8) What do the CSV and PDF exports include?

The exports summarize the computed results, including the entered polynomial, roots, factorization, remaining part, and graph interval. The PDF also captures the visible result section.