Difference of Cubes Identity with Factoring Examples
Difference of Cubes Identity expresses the subtraction of two perfect cubes as a product of two factors. The standard formula is a³ − b³ = (a − b)(a² + ab + b²). This page explains its expansion, factorisation method, sign pattern and numerical applications for quantitative aptitude.
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What Is the Difference of Cubes Identity?
Here, a and b may be numbers, variables or algebraic expressions. The first factor contains the difference, while the second factor contains the square of each term and their positive product. For example, x³ − 8 = x³ − 2³ = (x − 2)(x² + 2x + 4).
Difference of Cubes Identity Video Lessons
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Cube Identities: A Plus or Minus B
Learn the cube identities for (A+B)³ and (A−B)³, along with the factorisations of A³+B³ and A³−B³ for solving algebraic expressions.
Difference of Cubes Identity Formula & Tricks
Important Formulas
Use this identity when two cubes are subtracted.
On multiplication, the middle terms cancel: a²b − a²b and −ab² + ab².
This is the same identity applied to numerical values m and n.
Quick Tricks
For a³ − b³, the signs in the factors are minus, plus, plus: (a − b)(a² + ab + b²).
Rewrite each term as a cube before applying the identity. Common cubes include 1, 8, 27, 64, 125 and 216.
If a − b is easy to find, calculate the quadratic factor separately and multiply the two results.
Difference of Cubes Identity Concepts
Factorising an Algebraic Difference of Cubes
First identify the cube roots of the two terms. Then form the factors (a − b) and (a² + ab + b²). The terms may include variables, such as 8x³ = (2x)³.
Expanding a Product into a Difference of Cubes
Multiply a by the quadratic factor and then subtract b times the same factor: a(a² + ab + b²) − b(a² + ab + b²). The mixed terms cancel, leaving a³ − b³.
Applying the Identity to Numerical Values
This method reduces the subtraction of large cubes to multiplication of smaller expressions. It is especially convenient when the cube roots differ by a small number.
Sign Difference Between Sum and Difference of Cubes
Only the first factor follows the sign between the cubes; the quadratic factor uses the opposite sign for its middle term. Thus, a³ − b³ and a³ + b³ must not be factorised with the same pattern.
Difference of Cubes Identity Data Table
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Difference and Sum of Cubes Sign Patterns
The factor signs depend on whether the two cubes are subtracted or added.
| Expression | First factor | Quadratic factor |
|---|---|---|
| a³ − b³ | a − b | a² + ab + b² |
| a³ + b³ | a + b | a² − ab + b² |
Note: For the difference of cubes, all terms in the quadratic factor are positive.
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Difference of Cubes Identity: Revision Points
Recall the formula, factorisation method and sign rules for quick algebraic calculations.
- a³ − b³ = (a − b)(a² + ab + b²).
- Rewrite terms such as 8x³, 27y³ and 125 as (2x)³, (3y)³ and 5³.
- The difference-of-cubes sign pattern is minus, plus, plus.
- The quadratic factor is a² + ab + b², with no negative term.
- For verification, expand the product and cancel the two pairs of mixed terms.
- Do not confuse a³ − b³ with a³ + b³; the sum identity is (a + b)(a² − ab + b²).
Difference of Cubes Identity FAQs
What is the difference of cubes identity formula?
The formula is a³ − b³ = (a − b)(a² + ab + b²).
How do you factorise x³ − 64?
Since 64 = 4³, x³ − 64 = x³ − 4³ = (x − 4)(x² + 4x + 16).
What is the factorisation of 8a³ − 27b³?
8a³ − 27b³ = (2a)³ − (3b)³ = (2a − 3b)(4a² + 6ab + 9b²).
How can 18³ − 17³ be calculated using the identity?
18³ − 17³ = (18 − 17)(18² + 18×17 + 17²) = 1×919 = 919.
What is the difference between the sum and difference of cubes identities?
a³ − b³ = (a − b)(a² + ab + b²), whereas a³ + b³ = (a + b)(a² − ab + b²).
Why do the middle terms cancel during expansion?
In (a − b)(a² + ab + b²), the products −a²b and +a²b cancel, as do −ab² and +ab², leaving a³ − b³.