Difference of Cubes Calculator
Factor expressions in the form a³ - b³ using the formula (a - b)(a² + ab + b²). Get complete step-by-step solutions.
Enter Cube Root Values
Enter a and b for: (ax)³ - b³
Sign Pattern:
Binomial: (a - b) - Same sign as original
Trinomial: (a² + ab + b²) - All positive
Understanding Difference of Cubes
The difference of cubes is a special factoring pattern for expressions where two perfect cubes are subtracted. This formula is closely related to sum of cubes, but with different signs.
The Formula:
a³ - b³ = (a - b)(a² + ab + b²)
Comparison with Sum of Cubes:
Sum: a³ + b³ = (a + b)(a² - ab + b²)
Difference: a³ - b³ = (a - b)(a² + ab + b²)
Notice: The binomial mirrors the original sign, the trinomial has opposite middle sign
Common Examples:
x³ - 27 = (x - 3)(x² + 3x + 9)
x³ - 8 = (x - 2)(x² + 2x + 4)
8x³ - 1 = (2x - 1)(4x² + 2x + 1)
x³ - 64 = (x - 4)(x² + 4x + 16)
27x³ - 8 = (3x - 2)(9x² + 6x + 4)
Key Differences from Difference of Squares:
- Cubes produce a binomial AND a trinomial (not two binomials)
- The trinomial typically doesn't factor further
- All terms in the trinomial are positive
- Works with both variables and constants
Common Mistakes to Avoid:
- Confusing the signs between sum and difference of cubes
- Using difference of squares formula instead
- Making the middle term of the trinomial negative (it's positive!)
- Trying to factor the trinomial further (it's usually prime)
- Forgetting to check if the original terms are perfect cubes
Frequently Asked Questions
What is the difference of cubes formula?
The difference of cubes formula is a³ - b³ = (a - b)(a² + ab + b²). It factors a binomial where two perfect cubes are subtracted into a binomial times a trinomial.
How is difference of cubes different from sum of cubes?
The main difference is in the signs. Difference of cubes: (a - b)(a² + ab + b²) has all positive terms in the trinomial. Sum of cubes: (a + b)(a² - ab + b²) has a negative middle term in the trinomial.
Why are all terms in the trinomial positive?
This is a property of the difference of cubes formula. When you multiply (a - b)(a² + ab + b²) and simplify, you always get a³ - b³. The positive signs in the trinomial are essential for the formula to work correctly.
Can I factor the trinomial factor further?
In most cases, no. The trinomial (a² + ab + b²) resulting from difference of cubes is typically prime and cannot be factored further over real numbers. This is an important characteristic of this factoring pattern.
How do I remember which formula to use?
Look at the operation between the cubes. If it's subtraction (a³ - b³), use (a - b)(a² + ab + b²). If it's addition (a³ + b³), use (a + b)(a² - ab + b²). The binomial always matches the original operation.
What are common perfect cubes I should know?
Memorize: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. For variables: x³, 8x³ = (2x)³, 27x³ = (3x)³, 64x³ = (4x)³. Being familiar with these will help you quickly identify difference of cubes patterns.
Can I use this formula with negative numbers?
Yes! If you have something like x³ - (-8), this becomes x³ + 8, which is a sum of cubes. Also, (-2)³ = -8, so be careful with signs when identifying cube roots of negative numbers.
How do I verify my answer is correct?
Multiply your factors back together: (a - b)(a² + ab + b²). Use the distributive property to multiply each term in the binomial by each term in the trinomial. If you get a³ - b³, your factoring is correct.
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