About Factoring Calculator
This factoring calculator is free, needs no sign-up and runs in your browser, so nothing you type is uploaded. It factors polynomials in one variable and shows which method it used at each step: taking out the greatest common factor, the difference of squares, sums and differences of cubes, perfect square trinomials, factoring by grouping, and the rational root theorem for trinomials such as 2x² + 7x + 3.
Every result is written in KaTeX and followed by a check: the factors are multiplied back out and compared with what you typed. If the polynomial has no factors with rational coefficients, the calculator says it is irreducible instead of inventing an answer.
The second mode takes a whole number and lists every factor pair, all its factors and its prime factorization, which is what many "factor 36" homework questions ask for. The polynomial keypad is built in, so it works on a phone without a symbol keyboard.
How to use Factoring Calculator
- Pick Polynomial or Whole number.
- For a polynomial, type it using ^ for powers, for example x^3 + x^2 + 2x + 2, or use the keypad.
- Read the factored form at the top. It updates as you type.
- Open the steps to see each method applied, and the expansion check underneath.
- For a whole number, read the factor pairs, the full list of factors and the prime factorization.
Features
- Greatest common factor taken out first, including numbers and powers of x.
- Difference of squares, sum of cubes and difference of cubes recognised by pattern.
- Perfect square trinomials and general trinomials through the rational root theorem.
- Factoring by grouping for four-term polynomials.
- Repeated application, so x⁴ - 16 splits into (x - 2)(x + 2)(x² + 4).
- An expansion check that multiplies the factors back out.
- Whole-number mode with factor pairs, all factors and the prime factorization.
- Math keypad and live preview, usable on a phone.
The factoring patterns
Most school factoring is recognising one of a few patterns. The calculator tries them in order: common factor first, then the special products, then grouping and the rational root theorem.
a² - b² = (a - b)(a + b)
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
a² + 2ab + b² = (a + b)²
x² + (p + q)x + pq = (x + p)(x + q)Worked examples
2x² + 7x + 3 gives (2x + 1)(x + 3)
The rational root theorem finds the roots -1/2 and -3, which give the factors (2x + 1) and (x + 3).
4x² - 25 gives (2x - 5)(2x + 5)
A difference of squares: (2x)² - 5².
x³ + 8 gives (x + 2)(x^2 - 2x + 4)
A sum of cubes: x³ + 2³.
x³ + x² + 2x + 2 gives (x + 1)(x^2 + 2)
Group as x²(x + 1) + 2(x + 1), then take out the common (x + 1).
Factor pairs of 360 gives 2^3 × 3^2 × 5, 24 factors
Whole-number mode lists every pair such as 1 × 360, 2 × 180, 3 × 120 and so on up to 18 × 20.
Pick the method
| What you see | Method | Example |
|---|---|---|
| Every term shares a factor | Greatest common factor | 6x² + 9x = 3x(2x + 3) |
| Two squares subtracted | Difference of squares | x² - 9 = (x - 3)(x + 3) |
| Two cubes added or subtracted | Sum or difference of cubes | x³ - 27 = (x - 3)(x² + 3x + 9) |
| Three terms, a is 1 | Find two numbers with sum b and product c | x² + 5x + 6 = (x + 2)(x + 3) |
| Four terms | Grouping | x³ + x² + 2x + 2 = (x + 1)(x² + 2) |
Good to know
What this calculator does not do
It factors polynomials in one variable over the rational numbers. Expressions with two variables, such as 12x²y + 18xy², are not supported yet and give a message.
Repeated factors are found with a square-free step, so x⁴ + 4x² + 4 gives (x² + 2)². A polynomial of degree four to eight with no rational root is searched for a split into two integer polynomials of lower degree (the Kronecker method), so x⁴ + 4 gives (x² − 2x + 2)(x² + 2x + 2) and x⁴ + x² + 1 gives (x² + x + 1)(x² − x + 1). Above degree eight, or with very large coefficients, the search stops and the polynomial is reported as irreducible even if it splits.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The expressions you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- Always take out the greatest common factor first. It often turns a hard trinomial into an easy one.
- Multiply your factors back out to check them. The calculator does this for you under the steps.
- If a quadratic is irreducible over the rationals, its discriminant is not a perfect square. Try the quadratic formula calculator for its exact roots.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Expressions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Expressions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I factor a trinomial?
For x² + bx + c, find two numbers that multiply to c and add to b. When the leading coefficient is not 1, find the roots with the rational root theorem or the quadratic formula and build the factors from them. The calculator does this and shows the roots it found.
What is the greatest common factor?
It is the largest expression that divides every term. For 6x² + 9x it is 3x, so the expression becomes 3x(2x + 3).
Can it factor sums of squares?
A sum of squares such as x² + 9 does not factor over the real numbers, so the calculator reports it as irreducible.
Does it factor with more than one variable?
Not yet. It works on polynomials in one variable and says so when it sees two.
What are factor pairs?
Two whole numbers that multiply to the given number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4.
How do I know the answer is right?
Under the steps there is an expansion check that multiplies the factors back out. It should match the original polynomial.
Popular Factoring Calculator pages
- Factor Trinomials CalculatorType a trinomial and get its two factors with the working.
- Difference of Squares CalculatorType an expression like 4x² - 25 and see it factored as a difference of squares.
- Sum and Difference of Cubes CalculatorType a cubic like 27x³ + 8 and see it factored with the cube formulas.
- Factor by Grouping CalculatorType a four-term polynomial and see it factored by grouping.
- Perfect Square Trinomial CalculatorType a trinomial and see whether it is a perfect square.
- GCF of a Polynomial CalculatorType a polynomial and see its greatest common factor taken out.
- Factor Pairs CalculatorEnter a whole number to get all its factor pairs and factors.