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Cubes and Sum of Cubes
Cubes have their own identities. 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. Sum of cubes: 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). Notice the middle sign inside the trinomial is opposite to the original sum or difference.
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Cubes and Sum of Cubes
What is it?
Cubes have their own identities. 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. Sum of cubes: 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). Notice the middle sign inside the trinomial is opposite to the original sum or difference.
In simple words
- 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.
Important terms
(a + b)3: The expansions are (a + b)3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)3 = a3 - 3a2 b + 3ab2 - b3.
(a - b)3: The expansions are (a + b)3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)3 = a3 - 3a2 b + 3ab2 - b3.
a3 + b3 = (a + b)(a2 - ab + b2): Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2).
a3 - b3 = (a - b)(a2 + ab + b2): Difference of cubes: a3 - b3 = (a - b)(a2 + ab + b2).
-ab: In a^3 + b^3 the trinomial has -ab, not +ab.
What you should learn
- 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.
Recommended for Class 9
Full school-level explanation
Cubes have their own identities. 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. Sum of cubes: 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). Notice the middle sign inside the trinomial is opposite to the original sum or difference.
WHAT — Core idea
Cubes have their own identities. 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. Sum of cubes: 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). 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.
Remember
- (a + b)^3
- (a - b)^3
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
- -ab
- (a + b)^3
- a^3 + b^3
- (a + b)^3
Deeper understanding
Go beyond memorising Cubes and Sum of Cubes. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- (a + b)^3
- (a - b)^3
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
- (a + b)3: The expansions are (a + b)3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)3 = a3 - 3a2 b + 3ab2 - b3
- (a - b)3: The expansions are (a + b)3 = a3 + 3a2 b + 3ab2 + b3 and (a - b)3 = a3 - 3a2 b + 3ab2 - b3
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.
For long answers on Cubes and Sum of Cubes: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- (a + b)^3
- (a - b)^3
- a3 + b3 = (a + b)(a2 - ab + b2)
- a3 - b3 = (a - b)(a2 + ab + b2)
Important formulas
- (a + b)^3
- (a - b)^3
- a3 + b3 = (a + b)(a2 - ab + b2)
Common exam traps
- In a^3 + b^3 the trinomial has -ab, not +ab.
- Correction: Mixing that sign is the exam trap.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
In a^3 + b^3 the trinomial has -ab, not +ab.
- Mistake
Correction: Mixing that sign is the exam trap.
- Mistake
(a + b)^3 is not a^3 + b^3; the 3a^2 b and 3ab^2 terms must appear.
- Mistake
Factor cubes before you try random splitting.
- Mistake
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
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
(a + b)^3
(a - b)^3
a3 + b3 = (a + b)(a2 - ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Cubes have their own identities. 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. Sum of cubes: a^3 + b^3 =
a3 + b3 = (a + b)(a2 - ab + b2)
- Definition
- Cubes have their own identities. 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. Sum of cubes: a^3 + b^3 =
- Formula
- a3 + b3 = (a + b)(a2 - ab + b2)
- Critical warning
- In a^3 + b^3 the trinomial has -ab, not +ab.
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Exam Practice
Define
Q1. Define Cubes and Sum of Cubes.
Explain
Q2. Explain: WHAT — Core idea.
Example
Q3. Give one clear example of Cubes and Sum of Cubes and say why it fits.
Mistake
Q4. State one common mistake students make about Cubes and Sum of Cubes.
Exam note
Q5. Write a short exam note on Cubes and Sum of Cubes: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.