ICSE Class 8 Maths Chapter 15 Factorisation

Chapter 15: Factorisation

15.1 Review

Factors

Each of the numbers (constant or variable), which form a product is called a factor of the product.

(i) 5 and x are factors of the product 5x.

(ii) (2x - 5) and (3x + 2) are the factors of (2x - 5) (3x + 2).

Since, (2x - 5)(3x + 2) = 2x(3x + 2) - 5(3x + 2)

= 6x² + 4x - 15x - 10

= 6x² - 11x - 10

∴ 2x - 5 and 3x + 2 are the factors of 6x² - 11x - 10.

Test Yourself

1. Factors of xy are ......... and ..........

2. Factors of xy (4x + 7) are ........., ........ and ..................

3. Factors of xy(4x + 7)(x - 8) are ........, ............, ................. and ..................

4. Since, 4x² - 9 = (2x)² - (3)² = (2x + 3)(2x - 3); factors of 4x² - 9 are ........... and ..................

5. Since, x² - 5xy + 6y² = (x - 3y)(x - 2y); factors of x² - 5xy + 6y² are ............................................................................................

Teacher's Note

Factorisation is like breaking down a recipe into its basic ingredients - just as a cake is made of flour, eggs, and sugar, algebraic expressions can be broken down into their fundamental building blocks.

15.2 Factorisation

Factorisation means to find two or more expressions whose product is equal to the given expression.

15.3 Factorisation By Taking Out Common Factors

Steps:

1. Find by inspection, the largest monomial that will divide each term of the given polynomial completely.

2. Divide each term of the given polynomial by this monomial (factor) and enclose the quotient within brackets keeping this common monomial outside the bracket.

Example 1

Factorise:

(i) 5x² - 10x

(ii) 3x²y - 6xy² + 9xy

Solution:

(i) By inspection, we find that the largest monomial which divides each term of the given polynomial 5x² - 10x is 5x.

\[5x^2 - 10x = 5x\left(\frac{5x^2}{5x} - \frac{10x}{5x}\right)\]

= 5x (x - 2)

(ii)

\[3x^2y - 6xy^2 + 9xy = 3xy\left(\frac{3x^2y}{3xy} - \frac{6xy^2}{3xy} + \frac{9xy}{3xy}\right)\]

= 3xy (x - 2y + 3)

Example 2

Factorise:

(i) -10a⁴x² - 15a⁶x⁴ + 20a⁷x⁵

(ii) 2x(a + b) - 3y (a + b)

Solution:

(i)

\[-10a^4x^2 - 15a^6x^4 + 20a^7x^5 = -5a^4x^2\left(\frac{-10a^4x^2}{-5a^4x^2} - \frac{15a^6x^4}{-5a^4x^2} + \frac{20a^7x^5}{-5a^4x^2}\right)\]

= -5a⁴x²(2 + 3a²x² - 4a³x³)

(ii)

\[2x(a + b) - 3y(a + b) = (a + b)\left[\frac{2x(a+b)}{a+b} - \frac{3y(a+b)}{a+b}\right]\]

= (a + b)(2x - 3y)

Teacher's Note

Finding common factors is like identifying what ingredients appear in multiple recipes - just as you might notice butter is used in both cookies and cake, common terms appear across different parts of an algebraic expression.

Exercise 15 (A)

Factorise:

1. 15x + 5

2. a³ - a² + a

3. 3x² + 6x³

4. 4a² - 8ab

5. 2x³b² - 4x²b⁴

6. 15x⁴y³ - 20x³y

7. a³b - a²b² - b³

8. 6x²y + 9xy² + 4y³

9. 17a⁶b⁸ - 34a⁴b⁶ + 51a²b⁴

10. 3x⁵y - 27x⁴y² + 12x³y³

11. x²(a - b) - y²(a - b) + z²(a - b)

12. (x + y)(a + b) + (x - y)(a + b)

13. 2b (2a + b) - 3c (2a + b)

14. 12abc - 6a²b²c² + 3a³b³c³

15. 4x(3x - 2y) - 2y(3x - 2y)

16. (a + 2b) (3a + b) - (a + b) (a + 2b) + (a + 2b)²

17. 6xy(a² + b²) + 8yz(a² + b²) - 10xz(a² + b²)

15.4 Factorisation By Grouping

A given algebraic expression, containing an even number of terms may be resolved into factors, if its terms can be arranged in groups such that each group has a common factor.

Steps:

1. Arrange the terms of the given expression in suitable groups such that each group has a common factor.

2. Factorise each group.

3. Take out the factor which is common to each group.

Example 3

Factorise: ax - bx + ay - by

Solution:

ax - bx + ay - by

= (ax - bx) + (ay - by)

= x(a - b) + y(a - b)

= (a - b)(x + y)

Or ax - bx + ay - by

= ax + ay - bx - by

= a(x + y) - b(x + y)

= (x + y)(a - b)

Example 4

Factorise:

(i) y³ - 3y² + 2y - 6 - xy + 3x

(ii) a² - (b + 5) a + 5b

Solution:

(i) y³ - 3y² + 2y - 6 - xy + 3x = (y³ - 3y²) + (2y - 6) - (xy - 3x)

= y²(y - 3) + 2(y - 3) - x(y - 3)

= (y - 3)(y² + 2 - x)

(ii) a² - (b + 5) a + 5b = a² - ab - 5a + 5b

= (a² - ab) - (5a - 5b)

= a(a - b) - 5(a - b)

= (a - b) (a - 5)

Teacher's Note

Grouping terms in algebra is similar to sorting laundry - you group items by similarity (colors with colors, delicates with delicates) to process them more efficiently and reveal their underlying structure.

Exercise 15 (B)

Factorise:

1. a² + ax + ab + bx

2. a² - ab - ca + bc

3. ab - 2b + a² - 2a

4. a³ - a² + a - 1

5. 2a - 4b - xa + 2bx

6. xy - ay - ax + a² + bx - ab

7. 3x⁵ - 6x⁴ - 2x³ + 4x² + x - 2

8. -x²y - x + 3xy + 3

9. 6a² - 3a²b - bc² + 2c²

10. 3a²b - 12a² - 9b + 36

11. x² - (a - 3)x - 3a

12. x² - (b - 2)x - 2b

13. a(b - c) - d(c - b)

14. ab² - (a - c) b - c

15. (a² - b²) c + (b² - c²)a

16. a³ - a² - ab + a + b - 1

17. ab(c² + d²) - a²cd - b²cd

18. 2ab² - aby + 2cby - cy²

19. ax + 2bx + 3cx - 3a - 6b - 9c

20. 2ab²c - 2a + 3b⁶c - 3b - 4b²c² + 4c

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