Calculator inputs
Check your own factorization
Batch factoring mode
Recent calculations
Display options
Formula used
ab + ac = a(b + c)a² − b² = (a − b)(a + b)a² ± 2ab + b² = (a ± b)²a³ + b³ = (a + b)(a² − ab + b²)a³ − b³ = (a − b)(a² + ab + b²)aU² + bUV + cV²How to use this calculator
- Enter an expanded polynomial containing no more than two variables.
- Confirm the variable letters and choose the desired factoring domain.
- Select automatic detection or request a particular factoring method.
- Press Factor polynomial to view the result, method, and detailed steps.
- Use verification, copy, CSV, PDF, sharing, batch, or history tools as needed.
The calculator targets exact algebraic factorization. Real-domain output may use rounded roots when an exact rational factorization does not exist.
Example data
| Polynomial | Method | Expected factorization | Verification |
|---|---|---|---|
| 6x^2y + 9xy^2 | Greatest common factor | 3xy(2x + 3y) | Expansion matches |
| x^2 - y^2 | Difference of squares | (x - y)(x + y) | Expansion matches |
| x^2 + 2xy + y^2 | Perfect-square trinomial | (x + y)^2 | Expansion matches |
| 2x^2 + 5xy + 2y^2 | Homogeneous quadratic | (2x + y)(x + 2y) | Expansion matches |
| x^3 + x^2y - xy^2 - y^3 | Grouping | (x + y)^2(x - y) | Expansion matches |
| 4x^2 - 12xy + 9y^2 | Perfect-square trinomial | (2x - 3y)^2 | Expansion matches |
Understanding two-variable polynomial factoring
Factoring rewrites a polynomial as a product of simpler expressions. The product must expand back to the original polynomial. This calculator first combines like terms and identifies both variables.
A common monomial factor should be removed before other methods begin. Its coefficient is the rational greatest common factor of every coefficient. Its variable powers are the smallest powers appearing in all terms.
Special identities provide fast factorizations when term patterns match exactly. A difference of squares contains two square monomials with opposite signs. Perfect-square trinomials have a middle term equal to twice the product.
Many expressions behave like ordinary quadratics after substitution. For example, x⁴ + 5x²y² + 4y⁴ becomes U² + 5UV + 4V². Here U equals x², while V equals y².
Factoring by grouping works especially well with four terms. Two pairs are factored separately, then a repeated binomial is extracted. The calculator tests useful pairings instead of relying only on entered order.
The selected domain affects whether a result is considered complete. Integer factoring allows integer coefficients inside factors. Rational factoring permits fractions, while real factoring can introduce rounded irrational roots.
An irreducible result means no supported nonconstant factors were found in the selected domain. It does not mean the expression lacks complex factors. Changing the domain may therefore produce a different conclusion.
Verification protects against sign and coefficient errors. Every exact result is expanded internally and compared term by term. The checker also evaluates a factorization entered manually.
Use exact mode for assignments and symbolic work. Approximate mode is useful when real roots involve long decimals. Keep exact factors whenever later algebraic cancellation is expected.
Batch mode processes several exercises with identical settings. History stores recent expressions locally in the current browser. Export tools simplify reviewing, printing, and sharing saved results.
Frequently asked questions
What is a two-variable polynomial?
It is a finite sum of terms containing coefficients and nonnegative whole-number powers of two variables.
Does the calculator combine like terms?
Yes. Terms with identical exponent pairs are combined before the factoring methods are applied.
Can it factor fractional coefficients?
Yes. Fractional and terminating decimal coefficients are converted into exact rational values whenever possible.
What does irreducible mean?
It means no supported nonconstant factorization exists in the selected coefficient domain.
Can I use variables other than x and y?
Yes. Enter any two distinct single-letter variable names in the variable fields.
Why does real mode show decimals?
Some real factors contain irrational values, so the calculator displays controlled decimal approximations.
How is the result checked?
The factors are multiplied internally and the resulting coefficients are compared with the simplified input.
Can the calculator expand existing factors?
Yes. Enter them in the proposed-factor field and use the verification button.
Does batch mode use the same settings?
Yes. Every line uses the currently selected variables, domain, method, ordering, and output mode.