About Factor Trinomials Calculator
This factor trinomials calculator turns expressions of the form ax² + bx + c into a product of two brackets and shows how it found them. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
When a is 1 you look for two numbers that multiply to c and add to b. When a is not 1, as in 6x² - x - 2, that guesswork grows, so the calculator finds the rational roots first and builds the brackets from them. The page opens on 6x² - x - 2, which factors as (2x + 1)(3x - 2).
If the discriminant is not a perfect square, the trinomial has no rational factors and the calculator tells you it is irreducible.
How to use Factor Trinomials 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.
Trinomial factoring
If the roots are r and s, then ax² + bx + c = a(x - r)(x - s). Clear the fractions to get integer brackets.
x² + (p + q)x + pq = (x + p)(x + q)
ax² + bx + c = a(x - r)(x - s)Worked examples
6x² - x - 2 gives (2x + 1)(3x - 2)
The roots are 2/3 and -1/2.
x² - x - 6 gives (x + 2)(x - 3)
Two numbers that multiply to -6 and add to -1 are -3 and 2.
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 6x² - x - 2?
The roots are 2/3 and -1/2, so the factors are (3x - 2) and (2x + 1). Multiplying them gives 6x² - x - 2 again.
What if the trinomial will not factor?
Then its discriminant is not a perfect square and it has no factors with rational coefficients. Use the quadratic formula for the roots.
More Factoring Calculator pages
- Factoring CalculatorType a polynomial to factor it with every step, or switch to a whole number for its factor pairs.
- 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.