MathematicsChapter 4: Factorization and Algebraic Manipulation
Cubes and Sum of Cubes
Cubes have their own identities.
Learning objectives
- Explain Cubes And Sum Of Cubes in simple mathematical language.
- Read the comparison table and choose the correct rule or method.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
What this grid covers
Cubes have their own identities. The expansions are (a + b)^3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)^3 = a3 - 3a2 b + 3ab2 - b3. Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2). Difference of cubes: a3 - b3 = (a - b)(a2 + ab + b2). Notice the middle sign inside the trinomial is opposite to the original sum or difference.
Where / symbols
- (a + b)^3
- The expansions are (a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 and (a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3.
- (a - b)^3
- The expansions are (a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3 and (a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3.
- a^3 + b^3 = (a + b)(a^2 - ab + b^2)
- Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
- a^3 - b^3 = (a - b)(a^2 + ab + b^2)
- Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
- -ab
- In a^3 + b^3 the trinomial has -ab, not +ab.
- a^3 + b^3
- Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Relationship
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
WHAT — Core idea
Cubes have their own identities. The expansions are (a + b)^3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)^3 = a3 - 3a2 b + 3ab2 - b3. Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2). Difference of cubes: a3 - b3 = (a - b)(a2 + ab + b2). Notice the middle sign inside the trinomial is opposite to the original sum or difference.
WHY — Why it matters
This idea helps you recognise the right mathematical pattern, choose a safe method, and explain your working clearly.
HOW — Method and examples
(a + b)^3: a^3 + 3a^2 b + 3ab^2 + b^3 — All plus; coefficients 1, 3, 3, 1 (a - b)^3: a^3 - 3a^2 b + 3ab^2 - b^3 — Signs +, -, +, - a^3 + b^3: (a + b)(a^2 - ab + b^2) — Binomial plus; trinomial minus ab a^3 - b^3: (a - b)(a^2 + ab + b^2) — Binomial minus; trinomial plus ab 8x^3 - 27: (2x - 3)(4x^2 + 6x + 9) — Difference of cubes with a=2x, b=3 x^3 + 1: (x + 1)(x^2 - x + 1) — Sum of cubes, b = 1
HOW — Answer in an exam
In a^3 + b^3 the trinomial has -ab, not +ab. Mixing that sign is the exam trap. (a + b)^3 is not a^3 + b^3; the 3a^2 b and 3ab^2 terms must appear. Factor cubes before you try random splitting.
Keep expansion and factorisation as a matching pair:
| Identity | Right-hand side | Middle-sign hint |
|---|---|---|
| 1. (a + b)^3 | a^3 + 3a^2 b + 3ab^2 + b^3 | All plus; coefficients 1, 3, 3, 1 |
| 2. (a - b)^3 | a^3 - 3a^2 b + 3ab^2 - b^3 | Signs +, -, +, - |
| 3. a^3 + b^3 | (a + b)(a^2 - ab + b^2) | Binomial plus; trinomial minus ab |
| 4. a^3 - b^3 | (a - b)(a^2 + ab + b^2) | Binomial minus; trinomial plus ab |
| 5. 8x^3 - 27 | (2x - 3)(4x^2 + 6x + 9) | Difference of cubes with a=2x, b=3 |
| 6. x^3 + 1 | (x + 1)(x^2 - x + 1) | Sum of cubes, b = 1 |
In simple words
In a^3 + b^3 the trinomial has -ab, not +ab. Mixing that sign is the exam trap. (a + b)^3 is not a^3 + b^3; the 3a^2 b and 3ab^2 terms must appear. Factor cubes before you try random splitting.
Key points
- (a + b)^3
- (a - b)^3
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Cubes and Sum of Cubes
Sticky Notes · Quick Revision
WHAT
- Cubes have their own identities.
Key Points
- Cubes have their own identities.
- Keep expansion and factorisation as a matching pair: Review each row and apply its rule carefully.
- In a^3 + b^3 the trinomial has -ab, not +ab.
HOW
- (a + b)^3: a^3 + 3a^2 b + 3ab^2 + b^3 — All plus; coefficients 1, 3, 3, 1
- (a - b)^3: a^3 - 3a^2 b + 3ab^2 - b^3 — Signs +, -, +, -
- a^3 + b^3: (a + b)(a^2 - ab + b^2) — Binomial plus; trinomial minus ab
- a^3 - b^3: (a - b)(a^2 + ab + b^2) — Binomial minus; trinomial plus ab
- 8x^3 - 27: (2x - 3)(4x^2 + 6x + 9) — Difference of cubes with a=2x, b=3
Examples
- (a + b)^3: a^3 + 3a^2 b + 3ab^2 + b^3
- (a - b)^3: a^3 - 3a^2 b + 3ab^2 - b^3
- a^3 + b^3: (a + b)(a^2 - ab + b^2)
- a^3 - b^3: (a - b)(a^2 + ab + b^2)
- 8x^3 - 27: (2x - 3)(4x^2 + 6x + 9)
Common Mistake
- In a^3 + b^3 the trinomial has -ab, not +ab.
- Correction: Mixing that sign is the exam trap.
Exam Tip
- In a^3 + b^3 the trinomial has -ab, not +ab. Mixing that sign is the exam trap. (a + b)^3 is not a^3 + b^3; the 3a^2 b and 3ab^2 terms must appear. Factor cubes before you try random splitting.
60-second revision
- Cubes have their own identities.
- Keep expansion and factorisation as a matching pair: Review each row and apply its rule carefully.
- In a^3 + b^3 the trinomial has -ab, not +ab.
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