Factoring Calculator
Free factoring calculator for integers and polynomials. Prime factorization, GCD extraction, difference of squares, sum/difference of cubes, and rational roots with step-by-step explanations.
Factoring Calculator
This professional-grade factoring calculator helps students, educators, and professionals factor integers and polynomials quickly and accurately. It explains each step (GCD, special patterns, rational roots) and presents clean, reproducible results suitable for homework, teaching, or technical documentation.
Interactive Calculator
Results
Your factorization will appear here. The container reserves space to prevent layout shifts.
Authoritative Content Ecosystem
Data Source and Methodology
Authoritative data sources:
- OpenStax, Algebra and Trigonometry, Chapter 2: Factoring Polynomials (2016). https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry
- OpenStax, Prealgebra 2e, Section 5.4: Prime Factorization and the Fundamental Theorem of Arithmetic (2020). https://openstax.org/details/books/prealgebra-2e
All calculations are strictly based on the formulas and data provided by these sources.
The Formula Explained
Greatest common divisor (GCD) extraction:
Let P(x) have integer coefficients. If g = gcd of all coefficients, then $$P(x) = g \cdot Q(x)$$ where Q(x) has coprime integer coefficients.
Difference of squares:
$$a^2 - b^2 = (a - b)(a + b)$$
Sum/difference of cubes:
$$a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$$ $$a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$$
Quadratic factorization via discriminant:
For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $$ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$$
Rational Root Theorem:
If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
Glossary of Variables
- Input expression: The integer or polynomial to factor (variable x).
- GCD: Greatest common divisor of polynomial coefficients.
- Linear factor: A factor of the form \(ax + b\) with integers a, b.
- Irreducible quadratic: A factor \(ax^2 + bx + c\) with no rational roots.
- Prime factorization: Integer written as product of primes \(p_1^{e_1}\cdots p_k^{e_k}\).
Worked Example
How It Works: A Step-by-Step Example
Factor \(6x^2 - 5x - 6\).
- GCD of 6, -5, -6 is 1 ⇒ no common factor.
- By the Rational Root Theorem, try candidates ±1, ±2, ±3, ±6, ±1/2, ±3/2, ±1/3, ±2/3, ±1/6. Testing gives \(x = \frac{3}{2}\) as a root.
- Divide by \((2x - 3)\) to obtain \(3x + 2\).
- Therefore, \(6x^2 - 5x - 6 = (2x - 3)(3x + 2)\).
Frequently Asked Questions (FAQ)
Which inputs are supported?
Integers (for prime factorization) and single-variable polynomials in x with integer coefficients using ^ for powers.
Does it handle special patterns?
Yes. It detects GCDs, difference of squares, and sum/difference of cubes. It also performs rational-root deflation when applicable.
Over which field are factors computed?
All factoring is over the rationals (Q). Irreducible quadratics remain unfactored if they have no rational roots.
How large an integer can it factor?
It is tuned for up to about 10^12 using optimized trial division. Larger inputs might be slow; consider specialized tools for very large numbers.
Are decimals or other variables allowed?
No. Use integer coefficients and the variable x. Decimals and multiple variables are not supported for reliability.
Can I copy or share the result?
Yes. Use the Copy and Share buttons for fast collaboration and documentation.
Formula (LaTeX) + variables + units
','
P(x) = g \cdot Q(x)
a^2 - b^2 = (a - b)(a + b)
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)
a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)
ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).
Greatest common divisor (GCD) extraction: Let P(x) have integer coefficients. If g = gcd of all coefficients, then $P(x) = g \cdot Q(x)$ where Q(x) has coprime integer coefficients. Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ Sum/difference of cubes: $a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$ $a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$ Quadratic factorization via discriminant: For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$ Rational Root Theorem: If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
- T = property tax (annual or monthly depending on input) (currency)
- https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry - https://openstax.org/details/books/prealgebra-2e — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/prealgebra-2e
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.
Full original guide (expanded)
Skip to main contentSkip to main contentFactoring Calculator
This professional-grade factoring calculator helps students, educators, and professionals factor integers and polynomials quickly and accurately. It explains each step (GCD, special patterns, rational roots) and presents clean, reproducible results suitable for homework, teaching, or technical documentation.
Interactive Calculator
Results
Your factorization will appear here. The container reserves space to prevent layout shifts.
Authoritative Content Ecosystem
Data Source and Methodology
Authoritative data sources:
- OpenStax, Algebra and Trigonometry, Chapter 2: Factoring Polynomials (2016). https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry
- OpenStax, Prealgebra 2e, Section 5.4: Prime Factorization and the Fundamental Theorem of Arithmetic (2020). https://openstax.org/details/books/prealgebra-2e
All calculations are strictly based on the formulas and data provided by these sources.
The Formula Explained
Greatest common divisor (GCD) extraction:
Let P(x) have integer coefficients. If g = gcd of all coefficients, then $$P(x) = g \cdot Q(x)$$ where Q(x) has coprime integer coefficients.
Difference of squares:
$$a^2 - b^2 = (a - b)(a + b)$$
Sum/difference of cubes:
$$a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$$ $$a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$$
Quadratic factorization via discriminant:
For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $$ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$$
Rational Root Theorem:
If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
Glossary of Variables
- Input expression: The integer or polynomial to factor (variable x).
- GCD: Greatest common divisor of polynomial coefficients.
- Linear factor: A factor of the form \(ax + b\) with integers a, b.
- Irreducible quadratic: A factor \(ax^2 + bx + c\) with no rational roots.
- Prime factorization: Integer written as product of primes \(p_1^{e_1}\cdots p_k^{e_k}\).
Worked Example
How It Works: A Step-by-Step Example
Factor \(6x^2 - 5x - 6\).
- GCD of 6, -5, -6 is 1 ⇒ no common factor.
- By the Rational Root Theorem, try candidates ±1, ±2, ±3, ±6, ±1/2, ±3/2, ±1/3, ±2/3, ±1/6. Testing gives \(x = \frac{3}{2}\) as a root.
- Divide by \((2x - 3)\) to obtain \(3x + 2\).
- Therefore, \(6x^2 - 5x - 6 = (2x - 3)(3x + 2)\).
Frequently Asked Questions (FAQ)
Which inputs are supported?
Integers (for prime factorization) and single-variable polynomials in x with integer coefficients using ^ for powers.
Does it handle special patterns?
Yes. It detects GCDs, difference of squares, and sum/difference of cubes. It also performs rational-root deflation when applicable.
Over which field are factors computed?
All factoring is over the rationals (Q). Irreducible quadratics remain unfactored if they have no rational roots.
How large an integer can it factor?
It is tuned for up to about 10^12 using optimized trial division. Larger inputs might be slow; consider specialized tools for very large numbers.
Are decimals or other variables allowed?
No. Use integer coefficients and the variable x. Decimals and multiple variables are not supported for reliability.
Can I copy or share the result?
Yes. Use the Copy and Share buttons for fast collaboration and documentation.
Formula (LaTeX) + variables + units
','
P(x) = g \cdot Q(x)
a^2 - b^2 = (a - b)(a + b)
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)
a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)
ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).
Greatest common divisor (GCD) extraction: Let P(x) have integer coefficients. If g = gcd of all coefficients, then $P(x) = g \cdot Q(x)$ where Q(x) has coprime integer coefficients. Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ Sum/difference of cubes: $a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$ $a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$ Quadratic factorization via discriminant: For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$ Rational Root Theorem: If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
- T = property tax (annual or monthly depending on input) (currency)
- https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry - https://openstax.org/details/books/prealgebra-2e — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/prealgebra-2e
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.
Factoring Calculator
This professional-grade factoring calculator helps students, educators, and professionals factor integers and polynomials quickly and accurately. It explains each step (GCD, special patterns, rational roots) and presents clean, reproducible results suitable for homework, teaching, or technical documentation.
Interactive Calculator
Results
Your factorization will appear here. The container reserves space to prevent layout shifts.
Authoritative Content Ecosystem
Data Source and Methodology
Authoritative data sources:
- OpenStax, Algebra and Trigonometry, Chapter 2: Factoring Polynomials (2016). https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry
- OpenStax, Prealgebra 2e, Section 5.4: Prime Factorization and the Fundamental Theorem of Arithmetic (2020). https://openstax.org/details/books/prealgebra-2e
All calculations are strictly based on the formulas and data provided by these sources.
The Formula Explained
Greatest common divisor (GCD) extraction:
Let P(x) have integer coefficients. If g = gcd of all coefficients, then $$P(x) = g \cdot Q(x)$$ where Q(x) has coprime integer coefficients.
Difference of squares:
$$a^2 - b^2 = (a - b)(a + b)$$
Sum/difference of cubes:
$$a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$$ $$a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$$
Quadratic factorization via discriminant:
For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $$ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$$
Rational Root Theorem:
If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
Glossary of Variables
- Input expression: The integer or polynomial to factor (variable x).
- GCD: Greatest common divisor of polynomial coefficients.
- Linear factor: A factor of the form \(ax + b\) with integers a, b.
- Irreducible quadratic: A factor \(ax^2 + bx + c\) with no rational roots.
- Prime factorization: Integer written as product of primes \(p_1^{e_1}\cdots p_k^{e_k}\).
Worked Example
How It Works: A Step-by-Step Example
Factor \(6x^2 - 5x - 6\).
- GCD of 6, -5, -6 is 1 ⇒ no common factor.
- By the Rational Root Theorem, try candidates ±1, ±2, ±3, ±6, ±1/2, ±3/2, ±1/3, ±2/3, ±1/6. Testing gives \(x = \frac{3}{2}\) as a root.
- Divide by \((2x - 3)\) to obtain \(3x + 2\).
- Therefore, \(6x^2 - 5x - 6 = (2x - 3)(3x + 2)\).
Frequently Asked Questions (FAQ)
Which inputs are supported?
Integers (for prime factorization) and single-variable polynomials in x with integer coefficients using ^ for powers.
Does it handle special patterns?
Yes. It detects GCDs, difference of squares, and sum/difference of cubes. It also performs rational-root deflation when applicable.
Over which field are factors computed?
All factoring is over the rationals (Q). Irreducible quadratics remain unfactored if they have no rational roots.
How large an integer can it factor?
It is tuned for up to about 10^12 using optimized trial division. Larger inputs might be slow; consider specialized tools for very large numbers.
Are decimals or other variables allowed?
No. Use integer coefficients and the variable x. Decimals and multiple variables are not supported for reliability.
Can I copy or share the result?
Yes. Use the Copy and Share buttons for fast collaboration and documentation.
Formula (LaTeX) + variables + units
','
P(x) = g \cdot Q(x)
a^2 - b^2 = (a - b)(a + b)
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)
a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)
ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).
Greatest common divisor (GCD) extraction: Let P(x) have integer coefficients. If g = gcd of all coefficients, then $P(x) = g \cdot Q(x)$ where Q(x) has coprime integer coefficients. Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ Sum/difference of cubes: $a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$ $a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$ Quadratic factorization via discriminant: For \(ax^2 + bx + c\), if \(D = b^2 - 4ac\) is a perfect square then $ax^2 + bx + c = a\left(x - \frac{-b + \sqrt{D}}{2a}\right)\left(x - \frac{-b - \sqrt{D}}{2a}\right).$ Rational Root Theorem: If \(P(x)\) has integer coefficients and a rational root \(r = \frac{p}{q}\) in lowest terms, then \(p\) divides the constant term and \(q\) divides the leading coefficient. Dividing by \((q x - p)\) reduces the degree.
- T = property tax (annual or monthly depending on input) (currency)
- https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/subcategories/core-math-algebra-and-trigonometry - https://openstax.org/details/books/prealgebra-2e — openstax.org · Accessed 2026-01-19
https://openstax.org/details/books/prealgebra-2e
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.