About Sum and Difference of Cubes Calculator
This calculator factors a sum of cubes a³ + b³ and a difference of cubes a³ - b³ and shows the formula and the cube roots it used. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
Both terms must be perfect cubes. The page opens on 27x³ + 8, where 27x³ is (3x)³ and 8 is 2³, so a = 3x and b = 2. The result is (3x + 2)(9x² - 6x + 4).
The quadratic factor in each result does not factor further over the real numbers, so the calculator stops there. A common factor, as in 2x³ - 16, is taken out first.
How to use Sum and Difference of Cubes 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.
Cube formulas
The signs follow the mnemonic SOAP: same, opposite, always positive.
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)Worked examples
27x³ + 8 gives (3x + 2)(9x^2 - 6x + 4)
a = 3x, b = 2.
x³ - 27 gives (x - 3)(x^2 + 3x + 9)
a = x, b = 3.
2x³ - 16 gives 2(x - 2)(x^2 + 2x + 4)
Take out 2 first, leaving the difference of cubes x³ - 8.
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.
What is the formula for the sum of cubes?
a³ + b³ = (a + b)(a² - ab + b²). The difference is a³ - b³ = (a - b)(a² + ab + b²).
Why does the quadratic factor not split?
Its discriminant is always negative, so it has no real roots and cannot be factored over the real numbers.
More Factoring Calculator pages
- Factoring CalculatorType a polynomial to factor it with every step, or switch to a whole number for its factor pairs.
- 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.
- 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.