Class 8 Maths Chapter 6: We Distribute, Yet Things Multiply Algebra & Identities Explained

 Class 8 Maths – Chapter 6: We Distribute, Yet Things Multiply
(Ganita Prakash Part 1)

Introduction

This chapter explores the distributive property of multiplication over addition and subtraction. Students learn how to expand algebraic expressions, apply identities, and use distributivity for fast multiplication and geometric reasoning.

Basic Knowledge Required

• Multiplication and addition basics

• Properties of integers (positive and negative)

• Triangle congruence and geometry basics

• Algebraic notation and simplification

Important Definitions

Distributive Property

a(b + c) = ab + ac

Identity 1

(a + m)(b + n) = ab + mb + an + mn

Identity 1A

(a + b)² = a² + 2ab + b²

Identity 1B

(a − b)² = a² − 2ab + b²

Identity 1C

(a + b)(a − b) = a² − b²

Formulas / Concepts Used

• Expansion of brackets using distributive property

• Square of sum and difference identities

• Difference of squares identity

• Fast multiplication using distributivity (×11, ×101, ×1001)

• Geometric representation of algebraic identities

Solved Examples (Step‑by‑Step)

Example 1: Increase in product

23 × 27 → increase when 27 → 28

= 23 × 28 = 23 × 27 + 23

Increase = 23

Example 2: Both numbers increased by 1

(a + 1)(b + 1) = ab + a + b + 1

Example 3: One increased, one decreased

(a + 1)(b − 1) = ab + b − a − 1

Example 4: Expand (a + b)²

= a² + 2ab + b²

Example 5: Expand (a − b)²

= a² − 2ab + b²

Example 6: Expand (a + b)(a² + 2ab + b²)

= a³ + 3a²b + 3ab² + b³

Example 7: Fast multiplication

3874 × 11 = 38740 + 3874 = 42614

Rule: Add digits with neighbors.

Example 8: Geometric area

(60 + 5)² = 3600 + 25 + 600 = 4225

 FIGURE IT OUT — COMPLETE SOLUTIONS

1. Multiplication grid

Expressions: bd, etc. (Insert Image No.)

2. Expand products

(i) (3 + u)(v − 3) = 3v − 9 + uv − 3u

(ii) 3(15 + 6a) = 45 + 18a

(iii) (10a + b)(10c + d) = 100ac + 10ad + 10bc + bd

(iv) (3 − x)(x − 6) = 3x − 18 − x² + 6x = −x² + 9x − 18

(v) (−5a + b)(c + d) = −5ac − 5ad + bc + bd

(vi) (5 + z)(y + 9) = 5y + 45 + yz + 9z

3. Product unchanged

Examples: (2, 6), (3, 5), (4, 4)

4. Expand

(i) (a + ab − 3b²)(4 + b)

(ii) (4y + 7)(y + 11z − 3)

5. Pattern expansions

(i) (a − b)(a + b) = a² − b²

(ii) (a − b)(a² + ab + b²) = a³ − b³

(iii) (a − b)(a³ + a²b + ab² + b³) = a⁴ − b⁴

Next identity: (a − b)(aⁿ⁻¹ + … + bⁿ⁻¹) = aⁿ − bⁿ

6. Fast multiplication

94 × 11 = 1034

495 × 11 = 5445

3279 × 11 = 36069

4791256 × 11 = 52703816

7. Square identities

(i) (m + 3)² = m² + 6m + 9

(ii) (6 + p)² = p² + 12p + 36

(iii) (6x + 5)² = 36x² + 60x + 25

8. Difference of squares

(i) (b − 6)² = b² − 12b + 36

(ii) (−2a + 3)² = 4a² − 12a + 9

(iii) (7y − 47)² = 49y² − 658y + 2209

9. Pattern verification

2(a² + b²) = (a + b)² + (a − b)²

10. Calendar diagonals

Products differ by constant values 

 Conclusion

This chapter shows how distributive property and algebraic identities simplify multiplication, expansion, and geometric reasoning. These tools are powerful for solving problems quickly and efficiently.

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FAQs – Chapter 6: We Distribute, Yet Things Multiply

1. What is distributive property?

It states a(b + c) = ab + ac.

2. What is Identity 1?

(a + m)(b + n) = ab + mb + an + mn.

3. What is (a + b)²?

a² + 2ab + b².

4. What is (a − b)²?

a² − 2ab + b².

5. What is (a + b)(a − b)?

a² − b².

6. How does distributivity help in fast multiplication?

It allows breaking numbers into parts (e.g., ×11, ×101).

7. Expand (3 − x)(x − 6).

= −x² + 9x − 18.

8. Expand (10a + b)(10c + d).

= 100ac + 10ad + 10bc + bd.

9. Expand (a − b)(a² + ab + b²).

= a³ − b³.

10. What is Identity 1A?

(a + b)² = a² + 2ab + b².

11. What is Identity 1B?

(a − b)² = a² − 2ab + b².

12. What is Identity 1C?

(a + b)(a − b) = a² − b².

13. Expand (6x + 5)².

= 36x² + 60x + 25.

14. Expand (−2a + 3)².

= 4a² − 12a + 9.

15. What is the general identity for (a − b)(aⁿ⁻¹ + … + bⁿ⁻¹)?

= aⁿ − bⁿ.