Class 8 Maths We Distribute Yet Things Multiply Notes

Chapter 6: We Distribute, Yet Things Multiply

Overview:
This chapter introduces algebraic identities and distributive properties, showing how multiplication can be simplified using patterns and formulas. Students learn to expand, factorize, and simplify expressions efficiently. The chapter helps in understanding the power of algebraic distribution in calculations.

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Key Concepts

1. Distributive Property:
   • Formula: a(b + c) = ab + ac
   • This allows breaking complex multiplication into simpler steps.
   • Example: 3 × (4 + 5) = 3×4 + 3×5 = 12 + 15 = 27
2. Algebraic Identities:
   • (a + b)² = a² + 2ab + b²
   • (a − b)² = a² − 2ab + b²
   • (a + b)(a − b) = a² − b²
3. Factorization:
   • Expressing expressions as product of simpler expressions using identities.
   • Example: x² + 5x + 6 = (x + 2)(x + 3)
4. Using Identities to Multiply Numbers:
   • Example: Multiply 98 × 102 using (a − b)(a + b) = a² − b²
     • 98 × 102 = (100−2)(100+2) = 100² − 2² = 10000 − 4 = 9996
5. Patterns in Numbers:
   • Using distributive property and squares to simplify calculations.
   • Example: (x + 5)(x + 1) = x² + 6x + 5

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Important Points to Remember:

• Distributive property simplifies multiplication.
• Algebraic identities help expand and factorize expressions quickly.
• Factoring helps solve equations and simplify problems.
• Look for patterns in numbers to reduce calculation steps.

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Examples:

1. Using Distributive Property:
   • 7 × 23 = 7 × (20 + 3) = 7×20 + 7×3 = 140 + 21 = 161
2. Using Identities to Multiply:
   • 102 × 98 = (100 + 2)(100 − 2) = 100² − 2² = 10000 − 4 = 9996
3. Factorization:
   • x² + 5x + 6 = (x + 2)(x + 3)

Questions

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A. Very Short Answer Questions (1–10)

1. State the distributive property.
2. Expand: 5(x + 3)
3. Expand: 7(a − 4)
4. Factorize: 6x + 9
5. Expand: (x + 5)²
6. Expand: (y − 3)²
7. Multiply using identity: (a + b)(a − b)
8. Factorize: x² − 9
9. Expand: (2x + 3)(2x − 3)
10. Multiply 99 × 101 using algebraic identity.

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B. Short Answer Questions (11–25)

11. Expand: (x + 7)(x + 2)
12. Expand: (y + 5)(y − 2)
13. Factorize: x² + 10x + 25
14. Factorize: x² − 16
15. Simplify: 3(x + 4) + 2(x − 1)
16. Simplify: 5(a − 2) − 3(a + 1)
17. Multiply using distributive property: 8 × 27
18. Multiply using identity: 104 × 96
19. Expand: (3x + 2)²
20. Expand: (5y − 4)²
21. Factorize: x² + 7x + 12
22. Factorize: x² − 25
23. Factorize: 4x² − 9
24. Multiply using identity: (50 + 2)(50 − 2)
25. Simplify: 2(x + 3) + 3(x − 1)

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C. Word Problems / Application Questions (26–40)

26. Multiply 98 × 102 using algebraic identity.
27. Multiply 47 × 53 using identity (a + b)(a − b)
28. Expand: (x + 8)(x + 5)
29. Expand: (y − 7)(y + 4)
30. Factorize: x² + 12x + 36
31. Factorize: x² − 49
32. A rectangle has sides (x + 3) and (x + 7). Find area in expanded form.
33. Simplify: 5(x + 6) − 3(x − 2)
34. Expand: (2x + 5)(2x − 5)
35. Multiply 101 × 99 using identity
36. Factorize: 9x² − 16
37. Factorize: 16y² − 25
38. Simplify: 3(a + 4) + 2(a + 5)
39. Expand: (x + 4)² − (x − 4)²
40. A square has side (x + 5). Find the area in expanded form.

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D. Higher Order / Thinking Questions (41–50)

41. Simplify: (x + 3)(x + 7) − (x − 2)(x + 2)
42. Factorize: x² + 11x + 28
43. Expand: (3x + 4)(3x − 4)
44. Multiply 96 × 104 using identity
45. Factorize: 25x² − 36
46. Expand and simplify: (x + 2)(x + 3) + (x − 1)(x + 5)
47. Simplify: 2(x + 3)² − 2(x − 3)²
48. Factorize: x² + 13x + 40
49. Multiply using identity: (200 + 3)(200 − 3)
50. A rectangle has length (x + 6) and width (x + 2). Find area in simplified form.

Answers

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A. Very Short Answer Questions (1–10)

1. Distributive property: a(b + c) = ab + ac
2. 5(x + 3) = 5×x + 5×3 = 5x + 15
3. 7(a − 4) = 7a − 28
4. 6x + 9 = 3(2x + 3) ✅
5. (x + 5)² = x² + 10x + 25
6. (y − 3)² = y² − 6y + 9
7. (a + b)(a − b) = a² − b²
8. x² − 9 = (x − 3)(x + 3)
9. (2x + 3)(2x − 3) = (2x)² − 3² = 4x² − 9
10. 99 × 101 = (100 − 1)(100 + 1) = 100² − 1² = 10000 − 1 = 9999

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B. Short Answer Questions (11–25)

11. (x + 7)(x + 2) = x² + 2x + 7x + 14 = x² + 9x + 14
12. (y + 5)(y − 2) = y² − 2y + 5y − 10 = y² + 3y − 10
13. x² + 10x + 25 = (x + 5)²
14. x² − 16 = (x − 4)(x + 4)
15. 3(x + 4) + 2(x − 1) = 3x + 12 + 2x − 2 = 5x + 10
16. 5(a − 2) − 3(a + 1) = 5a − 10 − 3a − 3 = 2a − 13
17. 8 × 27 = 8 × (20 + 7) = 160 + 56 = 216
18. 104 × 96 = (100 + 4)(100 − 4) = 100² − 4² = 10000 − 16 = 9984
19. (3x + 2)² = 9x² + 12x + 4
20. (5y − 4)² = 25y² − 40y + 16
21. x² + 7x + 12 = (x + 3)(x + 4)
22. x² − 25 = (x − 5)(x + 5)
23. 4x² − 9 = (2x − 3)(2x + 3)
24. (50 + 2)(50 − 2) = 50² − 2² = 2500 − 4 = 2496
25. 2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3

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C. Word Problems / Application Questions (26–40)

26. 98 × 102 = (100 − 2)(100 + 2) = 100² − 2² = 10000 − 4 = 9996
27. 47 × 53 = (50 − 3)(50 + 3) = 50² − 3² = 2500 − 9 = 2491
28. (x + 8)(x + 5) = x² + 5x + 8x + 40 = x² + 13x + 40
29. (y − 7)(y + 4) = y² + 4y − 7y − 28 = y² − 3y − 28
30. x² + 12x + 36 = (x + 6)²
31. x² − 49 = (x − 7)(x + 7)
32. Rectangle sides (x + 3) and (x + 7) → Area = (x + 3)(x + 7) = x² + 10x + 21
33. 5(x + 6) − 3(x − 2) = 5x + 30 − 3x + 6 = 2x + 36
34. (2x + 5)(2x − 5) = 4x² − 25
35. 101 × 99 = (100 + 1)(100 − 1) = 100² − 1² = 10000 − 1 = 9999
36. 9x² − 16 = (3x − 4)(3x + 4)
37. 16y² − 25 = (4y − 5)(4y + 5)
38. 3(a + 4) + 2(a + 5) = 3a + 12 + 2a + 10 = 5a + 22
39. (x + 4)² − (x − 4)² = (x² + 8x +16) − (x² − 8x +16) = 16x
40. Square side (x + 5) → Area = (x + 5)² = x² + 10x + 25

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D. Higher Order / Thinking Questions (41–50)

41. (x + 3)(x + 7) − (x − 2)(x + 2) = (x² + 10x + 21) − (x² − 4) = 10x + 25
42. x² + 11x + 28 = (x + 4)(x + 7)
43. (3x + 4)(3x − 4) = 9x² − 16
44. 96 × 104 = (100 − 4)(100 + 4) = 10000 − 16 = 9984
45. 25x² − 36 = (5x − 6)(5x + 6)
46. (x + 2)(x + 3) + (x − 1)(x + 5) = (x² + 5x + 6) + (x² + 4x − 5) = 2x² + 9x + 1
47. 2(x + 3)² − 2(x − 3)² = 2(x² + 6x + 9 − x² + 6x − 9) = 2(12x) = 24x
48. x² + 13x + 40 = (x + 5)(x + 8)
49. (200 + 3)(200 − 3) = 200² − 3² = 40000 − 9 = 39991
50. Rectangle area: (x + 6)(x + 2) = x² + 8x + 12