1. Put terms in standard form
Combine like powers and arrange them from the highest exponent down to the constant. The parser does this expansion exactly.
(x − 1)(x − 2)(x − 3)
Completely factored over ℚ
Polynomial Factoring Calculator with Steps
See standard form, coefficient GCF, rational zeros, repeated factors, synthetic division, any irreducible remainder, and an exact multiplication check.
Enter a polynomial
Factored result
Factorization over the rational numbers
Standard formx³ − 6x² + 11x − 6
Coefficient GCF1
Rational zeros1, 2, 3
VerificationExact multiplication check passed
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Step-by-step factorization
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How to factor a polynomial
2. Extract the GCF
Factor out the greatest shared numerical coefficient and the lowest shared power of x. This keeps later factors simpler.
3. Test rational zeros
For primitive integer coefficients, possible rational zeros have the form ±p/q, where p divides the constant and q divides the leading coefficient.
4. Divide and repeat
When P(r)=0, the Factor Theorem says (x−r) is a factor. Exact synthetic division lowers the degree, exposing repeated or remaining factors.
Factor Theorem
P(r) = 0 ⇔ (x − r) is a factor of P(x)
“Completely factored” depends on the number system. This calculator works over the rational numbers ℚ. For example, x²−2 stays irreducible here because its real factors contain irrational coefficients.
Factoring examples
Difference of squares
x²−9 has zeros 3 and −3, so x²−9=(x−3)(x+3).
GCF and a zero factor
6x³+18x²−24x=6x(x²+3x−4)=6x(x−1)(x+4).
Repeated factor
x³−6x²+12x−8=(x−2)³. The root 2 has multiplicity 3.
Irreducible over ℚ
x²+1 has no rational zeros, so it remains one quadratic factor over ℚ even though it factors over the complex numbers.
Exact arithmetic, privacy, and limits
Parsing, expansion, rational-root tests, polynomial division, and verification use reduced BigInt fractions. Decimal entries are converted to exact fractions, so the displayed algebra does not depend on floating-point rounding.
The calculator supports one-variable polynomials through degree 8 and numeric literals up to 18 digits. It does not accept equations, radicals, functions, negative powers, or variables other than x. For high-degree polynomials, rational linear factors are exact; bounded quadratic-factor searching covers ordinary classroom-sized coefficients, while any result that cannot be certified is explicitly marked as a partial factorization. Last reviewed: 4 August 2026.
Polynomial factoring FAQs
How do you factor a polynomial?
Write the polynomial in standard form, extract its greatest common factor, look for identities or grouping, then test possible rational zeros and divide out each exact factor.
What number system does this calculator use?
It factors over the rational numbers. A quadratic whose roots are irrational or complex stays as an irreducible quadratic factor.
Can the calculator find repeated factors?
Yes. It divides out the same exact factor repeatedly and combines duplicates with a power, such as (x−2)³.
Can I enter a factored or partly factored expression?
Yes. Parentheses, implicit multiplication, and whole-number powers are accepted. The tool expands the entry exactly before factoring it again.
Why does a high-degree result say partially factored?
Every displayed factor is exact, but some high-degree polynomials require algorithms beyond rational-root and bounded quadratic-factor searches. The unresolved polynomial is preserved instead of being incorrectly called irreducible.
How is the answer checked?
The calculator multiplies the exact scalar and every displayed factor, then compares every rational coefficient with the expanded input.