Complete Solutions and Summary of Factorisation – NCERT Class 8 Mathematics Chapter 12

Explanations, solved examples, and exercises on factorisation of natural numbers and algebraic expressions, including methods using common factors, regrouping, identities, division of algebraic expressions, and practical techniques from NCERT Class 8 Mathematics Chapter 12.

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Factorisation - Complete Study Guide

Chapter Overview

12.1 Introduction to Factors

12.2 Methods of Factorisation

12.3 Division of Expressions

12.4 Polynomial Division

What You'll Learn

Factors of Natural Numbers

Understanding prime factors and expressing numbers as products.

Algebraic Factorisation

Methods like common factors, regrouping, identities, and more.

Division Techniques

Dividing monomials, polynomials by monomials or polynomials.

Irreducible Factors

Identifying factors that cannot be broken down further.

Historical Context

This chapter builds on Class VI and VII concepts, focusing on expressing algebraic expressions as products of irreducible factors. It covers systematic methods for factorisation and division, essential for solving equations and simplifying expressions in algebra.

Key Highlights

Factorisation is key in algebra for simplifying expressions. Methods include common factors, regrouping, identities like \((a + b)^2 = a^2 + 2ab + b^2\), and polynomial division where remainder is zero.

Comprehensive Chapter Summary

1. Introduction to Factorisation

The chapter begins with factors of natural numbers, like 30 = 2 × 3 × 5 in prime form. It extends to algebraic expressions, where terms like 5xy are products of irreducible factors 5 × x × y.

2. Factors of Algebraic Expressions

Prime vs. Irreducible Factors

In algebra, 'irreducible' replaces 'prime'. For example, 3x(x + 2) = 3 × x × (x + 2), all irreducible.

Key Formulas

Expressions like \(5xy + 10x = 5x(y + 2)\). Note: 1 is a factor but not shown unless needed.

Examples

12a²b + 15ab² = 3ab(4a + 5b); 10x² - 18x³ + 14x⁴ = 2x²(7x² - 9x + 5).

3. Methods of Factorisation

Common Factors Method

Write terms as irreducible factors, separate commons, and combine using distributive law. E.g., 2x + 4 = 2(x + 2).

Regrouping Terms

Group terms with common factors. E.g., 2xy + 2y + 3x + 3 = (x + 1)(2y + 3).

Using Identities

Use \((a + b)^2 = a^2 + 2ab + b^2\), etc. E.g., x² + 8x + 16 = (x + 4)².

Factors of Form (x + a)(x + b)

For x² + px + q, find a, b where ab = q, a + b = p. E.g., x² + 5x + 6 = (x + 2)(x + 3).

4. Division of Algebraic Expressions

Monomial by Monomial

E.g., 6x³ ÷ 2x = 3x². Cancel common factors.

5. Polynomial Division

By Monomial

Divide each term or factor out common. E.g., (4y³ + 5y² + 6y) ÷ 2y = 2y² + (5/2)y + 3.

Polynomial by Polynomial

Factorise and cancel commons. E.g., (7x² + 14x) ÷ (x + 2) = 7x.

6. Additional Formulas and Examples

More identities: a² - b² = (a + b)(a - b). Examples like m⁴ - 256 = (m - 4)(m + 4)(m² + 16).

Questions and Answers from Chapter

Short Questions

Q1. Find the common factors of 12x, 36.

Answer: 12.

Q2. Find the common factors of 2y, 22xy.

Answer: 2y.

Q3. Find the common factors of 14pq, 28p²q².

Answer: 14pq.

Q4. Find the common factors of 2x, 3x², 4.

Answer: 1.

Q5. Find the common factors of 6abc, 24ab², 12a²b.

Answer: 6ab.

Q6. Find the common factors of 16x³, -4x², 32x.

Answer: 4x.

Q7. Find the common factors of 10pq, 20qr, 30rp.

Answer: 10.

Q8. Find the common factors of 3x²y³, 10x³y², 6x²y²z.

Answer: x²y².

Q9. Factorise 7x - 42.

Answer: 7(x - 6).

Q10. Factorise 6p - 12q.

Answer: 6(p - 2q).

Q11. Factorise 7a² + 14a.

Answer: 7a(a + 2).

Q12. Factorise -16z + 20z³.

Answer: 4z(-4 + 5z²).

Q13. Factorise 20l²m + 30alm.

Answer: 10lm(2l + 3a).

Q14. Factorise 5x²y - 15xy².

Answer: 5xy(x - 3y).

Q15. Factorise 10a² - 15b² + 20c².

Answer: 5(2a² - 3b² + 4c²).

Medium Questions

Q1. Factorise x² + xy + 8x + 8y.

Answer: Group as (x² + xy) + (8x + 8y) = x(x + y) + 8(x + y) = (x + y)(x + 8). (3 marks)

Q2. Factorise 15xy - 6x + 5y - 2.

Answer: Group as (15xy - 6x) + (5y - 2) = 3x(5y - 2) + 1(5y - 2) = (5y - 2)(3x + 1). (3 marks)

Q3. Factorise ax + bx - ay - by.

Answer: Group as (ax - ay) + (bx - by) = a(x - y) + b(x - y) = (x - y)(a + b). (3 marks)

Q4. Factorise 15pq + 15 + 9q + 25p.

Answer: Regroup as (15pq + 25p) + (15 + 9q) = 5p(3q + 5) + 3(5 + 3q) = (3q + 5)(5p + 3). (3 marks)

Q5. Factorise z - 7 + 7xy - xyz.

Answer: Group as (z - xyz) + (-7 + 7xy) = z(1 - xy) - 7(1 - xy) = (1 - xy)(z - 7). (3 marks)

Q6. Factorise a² + 8a + 16.

Answer: Recognize as (a + 4)² since a² + 2·a·4 + 4². (3 marks)

Q7. Factorise p² - 10p + 25.

Answer: (p - 5)² = p² - 2·p·5 + 5². (3 marks)

Q8. Factorise 25m² + 30m + 9.

Answer: (5m + 3)² = (5m)² + 2·5m·3 + 3². (3 marks)

Q9. Factorise 49y² + 84yz + 36z².

Answer: (7y + 6z)² = (7y)² + 2·7y·6z + (6z)². (3 marks)

Q10. Factorise 4x² - 8x + 4.

Answer: 4(x² - 2x + 1) = 4(x - 1)². (3 marks)

Q11. Factorise 121b² - 88bc + 16c².

Answer: (11b - 4c)² = (11b)² - 2·11b·4c + (4c)². (3 marks)

Q12. Factorise (l + m)² - 4lm.

Answer: Expand: l² + 2lm + m² - 4lm = l² - 2lm + m² = (l - m)². (3 marks)

Q13. Factorise a⁴ + 2a²b² + b⁴.

Answer: (a² + b²)² = a⁴ + 2a²b² + b⁴. (3 marks)

Q14. Factorise 4p² - 9q².

Answer: (2p - 3q)(2p + 3q). (3 marks)

Q15. Factorise 63a² - 112b².

Answer: 7(9a² - 16b²) = 7(3a - 4b)(3a + 4b). (3 marks)

Long Questions

Q1. Factorise -4a² + 4ab - 4ca and explain the process.

Answer: Factor out -4: -4(a² - ab + ca). Now, group a² - ab + ca. Regroup as a(a - b) + c(a). No common factor directly. Alternatively, think of it as -4a(a - b + c). Wait, let's check: Original is -4a² + 4ab - 4ca = -4a(a - b + c). Yes, because -4a·a = -4a², -4a·(-b) = 4ab, -4a·c = -4ac. Detailed process: Write each term: -4×a×a + 4×a×b - 4×c×a = -4a(a - b + c). Factors are -4a and (a - b + c).

Q2. Factorise x²yz + xy²z + xyz² and explain using common factors.

Answer: Each term has xyz common: x²yz = x·x·y·z, xy²z = x·y·y·z, xyz² = x·y·z·z. Common is x y z. So xyz(x + y + z). Detailed: Factor xyz out: xyz(x + y + z). Verification: xyz·x = x²yz, xyz·y = xy²z, xyz·z = xyz².

Q3. Factorise ax²y + bxy² + cxyz and explain the method.

Answer: Common factors: ax²y = a x x y, bxy² = b x y y, cxyz = c x y z. Common is x y. So xy(a x + b y + c z). Detailed process: Separate xy from each: xy(a x + b y + c z).

Q4. Factorise 49x² - 36 and explain using identity.

Answer: This is a² - b² where a = 7x, b = 6. So (7x - 6)(7x + 6). Detailed: Identity III: a² - b² = (a - b)(a + b). Here (7x)² - 6².

Q5. Factorise 16x⁵ - 144x³ and explain.

Answer: Factor 16x³(x² - 9) = 16x³(x - 3)(x + 3). Detailed: Common 16x³, then x² - 9 = (x - 3)(x + 3).

Q6. Factorise (l + m)² - (l - m)² and explain.

Answer: This is a² - b² where a = l + m, b = l - m. So [(l + m) - (l - m)] [(l + m) + (l - m)] = (2m)(2l) = 4lm. Detailed expansion: l² + 2lm + m² - (l² - 2lm + m²) = 4lm.

Q7. Factorise 9x²y² - 16 and explain.

Answer: (3xy)² - 4² = (3xy - 4)(3xy + 4). Detailed using identity III.

Q8. Factorise (x² - 2xy + y²) - z² and explain.

Answer: (x - y)² - z² = [(x - y) - z][(x - y) + z] = (x - y - z)(x - y + z). Detailed: First recognize as difference of squares.

Q9. Factorise ax² + bx and explain.

Answer: x(ax + b). Detailed: Common x.

Q10. Factorise 7p² + 21q² and explain.

Answer: 7(p² + 3q²). Detailed: Common 7.

Q11. Factorise 2x³ + 2xy² + 2xz² and explain.

Answer: 2x(x² + y² + z²). Detailed: Common 2x.

Q12. Factorise am² + bm² + bn² + an² and explain.

Answer: m²(a + b) + n²(a + b) = (a + b)(m² + n²). Detailed regrouping.

Q13. Factorise (lm + l) + m + 1 and explain.

Answer: l(m + 1) + 1(m + 1) = (l + 1)(m + 1). Detailed regrouping.

Q14. Factorise y(y + z) + 9(y + z) and explain.

Answer: (y + 9)(y + z). Detailed common (y + z).

Q15. Factorise 5y² - 20y - 8z + 2yz and explain.

Answer: 5y(y - 4) + 2z(y - 4) = (y - 4)(5y + 2z). Detailed regrouping.

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Complete Solutions and Summary of Factorisation – NCERT Class 8 Mathematics Chapter 12

Explanations, solved examples, and exercises on factorisation of natural numbers and algebraic expressions, including methods using common factors, regrouping, identities, division of algebraic expressions, and practical techniques from NCERT Class 8 Mathematics Chapter 12.

Factorisation

Chapter Overview

What You'll Learn

Factors of Natural Numbers

Algebraic Factorisation

Division Techniques

Irreducible Factors

Historical Context

Key Highlights

Comprehensive Chapter Summary

1. Introduction to Factorisation

2. Factors of Algebraic Expressions

Prime vs. Irreducible Factors

Key Formulas

Examples

3. Methods of Factorisation

Common Factors Method

Regrouping Terms

Using Identities

Factors of Form (x + a)(x + b)

4. Division of Algebraic Expressions

Monomial by Monomial

5. Polynomial Division

By Monomial

Polynomial by Polynomial

6. Additional Formulas and Examples

Key Concepts and Definitions

Factorisation

Irreducible Factor

Common Factors Method

Regrouping

Identities

Division

Important Facts and Figures

Questions and Answers from Chapter

Short Questions

Q1. Find the common factors of 12x, 36.

Q2. Find the common factors of 2y, 22xy.

Q3. Find the common factors of 14pq, 28p²q².

Q4. Find the common factors of 2x, 3x², 4.

Q5. Find the common factors of 6abc, 24ab², 12a²b.

Q6. Find the common factors of 16x³, -4x², 32x.

Q7. Find the common factors of 10pq, 20qr, 30rp.

Q8. Find the common factors of 3x²y³, 10x³y², 6x²y²z.

Q9. Factorise 7x - 42.

Q10. Factorise 6p - 12q.

Q11. Factorise 7a² + 14a.

Q12. Factorise -16z + 20z³.

Q13. Factorise 20l²m + 30alm.

Q14. Factorise 5x²y - 15xy².

Q15. Factorise 10a² - 15b² + 20c².

Medium Questions

Q1. Factorise x² + xy + 8x + 8y.

Q2. Factorise 15xy - 6x + 5y - 2.

Q3. Factorise ax + bx - ay - by.

Q4. Factorise 15pq + 15 + 9q + 25p.

Q5. Factorise z - 7 + 7xy - xyz.

Q6. Factorise a² + 8a + 16.

Q7. Factorise p² - 10p + 25.

Q8. Factorise 25m² + 30m + 9.

Q9. Factorise 49y² + 84yz + 36z².

Q10. Factorise 4x² - 8x + 4.

Q11. Factorise 121b² - 88bc + 16c².

Q12. Factorise (l + m)² - 4lm.

Q13. Factorise a⁴ + 2a²b² + b⁴.

Q14. Factorise 4p² - 9q².

Q15. Factorise 63a² - 112b².

Long Questions

Q1. Factorise -4a² + 4ab - 4ca and explain the process.

Q2. Factorise x²yz + xy²z + xyz² and explain using common factors.

Q3. Factorise ax²y + bxy² + cxyz and explain the method.

Q4. Factorise 49x² - 36 and explain using identity.

Q5. Factorise 16x⁵ - 144x³ and explain.

Q6. Factorise (l + m)² - (l - m)² and explain.

Q7. Factorise 9x²y² - 16 and explain.

Q8. Factorise (x² - 2xy + y²) - z² and explain.

Q9. Factorise ax² + bx and explain.

Q10. Factorise 7p² + 21q² and explain.