Sum of Cubes Identity with Factorisation Examples
Sum of Cubes Identity expresses the sum of two cubes as a product of three factors. The standard formula is a³ + b³ = (a + b)(a² − ab + b²). This page explains its derivation, factorisation method, related identities, sign rules and numerical examples used in quantitative aptitude.
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What Is the Sum of Cubes Identity?
For a³ + b³, the first factor is the sum of the bases, a + b. The second factor has a² and b² as positive terms, while the middle term is −ab. For example, x³ + 8 = x³ + 2³ = (x + 2)(x² − 2x + 4).
Sum 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.
Sum of Cubes Identity Formula & Tricks
Important Formulas
Use this formula when both terms are perfect cubes and are added.
On multiplying, the terms −a²b and +ab² cancel with +a²b and −ab².
If a + b and ab are known, factor out a + b to calculate the sum of cubes quickly.
Quick Tricks
For the sum of cubes, the factors are Sum × (Square − Product + Square). The signs are +, −, + inside the quadratic factor.
Rewrite numerical terms as cubes before applying the identity. Common cubes include 1, 8, 27, 64, 125 and 216.
Apply a³ + b³ = (a + b)³ − 3ab(a + b) instead of finding a and b separately.
Sum of Cubes Identity Concepts
Factorising a Sum of Two Cubes
First identify the cube roots of the two terms. Place their sum in the first factor. In the second factor, square the first root, subtract the product of the roots, and add the square of the second root.
Evaluating a Sum of Cubes from a Sum and Product
This follows from expanding (a + b)³ = a³ + 3a²b + 3ab² + b³ and combining the middle terms as 3ab(a + b).
Difference Between Sum and Difference of Cubes
The two identities are a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). The sign of the middle term changes when the original operation changes.
Applying the Identity to Algebraic Expressions
For k³a³, the cube root is ka. Thus, 8x³ + 27y³ becomes (2x)³ + (3y)³, not x³ + y³. Apply the identity only after rewriting both terms correctly.
Sum of Cubes Identity Data Table
Use the structured data below while following the explanation and examples.
Sum and Difference of Cubes
Comparison of the two standard cube factorisation identities.
| Expression | First factor | Second factor |
|---|---|---|
| a³ + b³ | a + b | a² − ab + b² |
| a³ − b³ | a − b | a² + ab + b² |
Note: The middle-term sign in the second factor is opposite to the sign between the original cubes.
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Sum of Cubes Identity Revision Points
Use these direct rules to revise factorisation and calculation methods.
- a³ + b³ = (a + b)(a² − ab + b²).
- The second factor follows the pattern square of the first base − product of the bases + square of the second base.
- Rewrite coefficients and variables as complete cubes before factorisation.
- a³ + b³ = (a + b)³ − 3ab(a + b).
- For a³ − b³, use (a − b)(a² + ab + b²), not the sum-of-cubes formula.
- Verify a factorisation by multiplying the two factors and checking cancellation of the mixed terms.
Sum of Cubes Identity FAQs
What is the sum of cubes identity formula?
The formula is a³ + b³ = (a + b)(a² − ab + b²).
How do you factor x³ + 8?
Since 8 = 2³, x³ + 8 = x³ + 2³ = (x + 2)(x² − 2x + 4).
What is the sum of cubes identity when a + b and ab are given?
Use a³ + b³ = (a + b)³ − 3ab(a + b). For a + b = 5 and ab = 6, the result is 125 − 90 = 35.
What is the difference between a³ + b³ and a³ − b³ factorisation?
a³ + b³ = (a + b)(a² − ab + b²), whereas a³ − b³ = (a − b)(a² + ab + b²). The middle-term sign changes.
How do you factor 64p³ + 125q³?
Write the terms as (4p)³ + (5q)³. Therefore, 64p³ + 125q³ = (4p + 5q)(16p² − 20pq + 25q²).
Can the sum of two cubes be divided by a + b?
Yes. Since a³ + b³ = (a + b)(a² − ab + b²), a + b is a factor of a³ + b³.