LESSON NOTES 2021 2022 Subject: Mathematics Chapter: 15 ...

LESSON NOTES 2021 – 2022

Subject: Mathematics Chapter: 15 Factorisation

Grade: VIII

NOTE: THE CONCEPT MAP AND ALL THE EXERCISE QUESTIONS TO BE SOLVED IN NOTEBOOK. Concept Map:

Factors: A factor is an exact divisor of that number. e.g.: 1, 2, 4 are the factors of 4.

1 is the factor of every number.

Every number is a factor of itself.

Exercise 15A

1. Find the greatest common factor of the given monomials.

i) 3𝑚, 6𝑛

Greatest common factor: 3

ii) 𝑎𝑥 − 5𝑎𝑏

Greatest common factor: 𝑎

iii) 18𝑥7, −9𝑥3

Greatest common factor: 9𝑥3

Algebraic Expression Factors

Factorization

Multiplication



Grade: VIII

iv) 10𝑥𝑦, 15𝑥2𝑦3

Greatest common factor: 5𝑥𝑦

v) 38𝑎3𝑥5𝑦, 15𝑥2𝑦3

Greatest common factor: 19𝑎3𝑥3

vi) 9𝑥𝑦2, −6𝑥𝑦, −3𝑥

Greatest common factor: 3𝑥

vii) 14𝑥3𝑦3, 21𝑥5𝑦2, 35𝑥6𝑦6

Greatest common factor: 7𝑥3𝑦2

viii) 𝑝2𝑞8𝑟3, −𝑝6𝑞7𝑟3, 𝑝𝑞5𝑟2

Greatest common factor: 𝑝𝑞5𝑟2

ix) 24𝑎2𝑏, −16𝑎𝑏𝑐, 32𝑎2𝑏2𝑐3, 36𝑎𝑏2𝑐2

Greatest common factor: 4𝑎𝑏

Factorise the following expressions.

2. 𝑎2 − 𝑎𝑥

=𝑎(𝑎 − 𝑥)

3. 3𝑎2 − 9

=3(𝑎2 − 3)

4. 5𝑎𝑥 − 5𝑎2𝑥2

=5𝑎𝑥(1 − 𝑎𝑥)

5. 𝜋𝑅2 − 𝜋𝑟2

=𝜋(𝑅2 − 𝑟2)

6. 10𝑥𝑦 − 15𝑥2𝑦2

=5𝑥𝑦(2 − 3𝑥𝑦)

7. 8𝑎4𝑏2𝑐3 + 12𝑎2𝑏𝑐3

=4𝑎2𝑏𝑐3(2𝑎2𝑏 + 3)

8. 𝑥2 − 𝑥𝑦 + 𝑥𝑦2

=𝑥(𝑥 − 𝑦 + 𝑦2)

9. 5𝑥4 − 10𝑎2𝑥2 − 15𝑎2𝑥3

=5𝑥2(𝑥2 − 2𝑎2𝑥2 − 3𝑎2𝑥)



Grade: VIII

10. 3𝑥3𝑦2 + 9𝑥2𝑦 + 12𝑥𝑦

=3𝑥𝑦(𝑥2𝑦 + 3𝑥 + 4)

11. 60𝑥2𝑦 + 25𝑦2 + 20𝑦2𝑧

Refer que, no. 10

12. 𝑎𝑥3𝑦2 + 𝑏𝑥2𝑦 + 𝑐𝑥𝑦𝑧

Refer que, no. 10

13. 8𝑥4 𝑦 + 2𝑥𝑦4 − 24𝑥2𝑦3 + 18𝑥3𝑦2

Refer que, no. 10

14. 3𝑥(𝑎 − 𝑏) − 6𝑦(𝑎 − 𝑏)

=3(𝑎 − 𝑏)(𝑥 − 2𝑦)

15. (𝑥 − 𝑦)2 − 2(𝑥 − 𝑦)

=(𝑥 − 𝑦)(𝑥 − 𝑦 − 2)

16. −3(𝑥 − 3𝑦) + 6(𝑥 − 3𝑦)2

=−3(𝑥 − 3𝑦)(1 − 2(𝑥 − 3𝑦))

=−3(𝑥 − 3𝑦)(1 − 2𝑥 + 6𝑦)

17. (𝑥 + 3)3 − 2(𝑥 = 3)2 + 3(𝑥3)

=(𝑥 + 3)[(𝑥 + 3)2 − 2(𝑥 + 3) + 3]

=(𝑥 + 3)[𝑥2 + 6𝑥 + 9 − 2𝑥 − 6 + 3]

=(𝑥 + 3)(𝑥2 + 4𝑥 + 6)

18. 𝑥(𝑎 + 𝑏)4 + 𝑦(𝑎 + 𝑏)3 + 𝑧(𝑎 + 𝑏)2

=(𝑎 + 𝑏)2 [ 𝑥(𝑎 + 𝑏)2 + 𝑦(𝑎 + 𝑏) + 𝑧]

=(𝑎 + 𝑏)2 [ 𝑥( 𝑎2 + 2𝑎𝑏 + 𝑏2) + 𝑦𝑎 + 𝑦𝑏 + 𝑧]

=(𝑎 + 𝑏)2 [ 𝑥𝑎2 + 2𝑥𝑎𝑏 + 𝑥𝑏2 + 𝑦𝑎 + 𝑦𝑏 + 𝑧]

19. 25(𝑥 − 𝑦)2 − 15(𝑥 − 𝑦) − 𝑥 + 𝑦

Refer Que 18

20. 𝑥2 + 𝑥𝑦 + 𝑥𝑧 + 𝑦𝑧

=𝑥(𝑥 + 𝑦) + 𝑧(𝑥 + 𝑦)

=(𝑥 + 𝑦)(𝑥 + 𝑧)

21. 𝑎2 − 𝑎𝑐 + 𝑎𝑏 − 𝑏𝑐

=Refer Que 20



Grade: VIII

22. 𝑎𝑥 − 𝑏𝑥 − 𝑎𝑧 + 𝑏𝑧

=𝑥(𝑎 − 𝑏) − 𝑧(𝑎 − 𝑏)

=(𝑎 − 𝑏)(𝑥 − 𝑧)

23. 4𝑥𝑦 − 𝑦2 + 8𝑥𝑧 − 2𝑦𝑧

=𝑦(4𝑥 − 𝑦) + 2𝑧(4𝑥 − 𝑦)

=(4𝑥 − 𝑦)(𝑦 + 2𝑧)

24. 𝑏𝑥 + 2𝑥𝑧 + 2𝑏𝑦 + 4𝑦𝑧

=𝑥(𝑏 + 2𝑧) + 2𝑦(𝑏 = 2𝑧)

=(𝑏 + 2𝑧)(𝑥 + 2𝑦)

25. 2𝑥2 + 3𝑥𝑦 − 2𝑥𝑧 − 3𝑦𝑧

Refer Que 20

26. 𝑝2𝑥 − 𝑝𝑞𝑦 − 2𝑝𝑥 − 2𝑞𝑦

=Refer Que 20

27. 𝑎𝑐2 − (𝑎 − 𝑏)𝑐 − 𝑏

=𝑎𝑐2 − 𝑎𝑐 + 𝑏𝑐 − 𝑏

=𝑎𝑐(𝑐 − 1) + 𝑏(𝑐 − 1)

=(𝑎𝑐 + 𝑏)(𝑐 − 1)

28. 𝑥3 − 𝑥2 + 𝑥 − 1

=𝑥2(𝑥 − 1) + 1(𝑥 − 1)

=(𝑥 − 1)(𝑥2 + 1)

29. 𝑥3 + 2𝑥 − 2𝑥2 − 4

=𝑥(𝑥2 + 2) − 2(𝑥2 + 2)

=(𝑥2 + 2) (x-2)

30. 𝑎𝑏2 − 𝑏𝑐2 − 𝑎𝑏 + 𝑐2

= 𝑏(𝑎𝑏 − 𝑐2) − 1(𝑎𝑏 − 𝑐2)

= (𝑎𝑏 − 𝑐2)(𝑏 − 1)

31. 8 − 4𝑥 − 2𝑥3 + 𝑥4

= 4(2 − 𝑥) − 𝑐𝑥3(2 − 𝑥)

= (4 − 𝑥3)(2 − 𝑥)



Grade: VIII

32. 𝑥3 − 2𝑥𝑦3 − 4𝑦3 + 2𝑥2𝑦

=𝑥(𝑥2 − 2𝑦2) + 2𝑦 (−2𝑦2 + 𝑥2)

=(𝑥2 − 2𝑦2)(𝑥 + 2𝑦)

33. 𝑎𝑥𝑦 + 𝑏𝑐𝑥𝑦 − 𝑎𝑧 − 𝑏𝑐𝑧

Refer Que 22

34. 2𝑎𝑥2 + 3𝑎𝑥𝑦 − 2𝑏𝑥𝑦 − 3𝑏𝑦2

Refer Que 25

35. 𝑎𝑥 + 𝑏𝑥 − 𝑎𝑦 − 𝑏𝑦 + 𝑎𝑧 + 𝑏𝑧

=𝑥(𝑎 + 𝑏) − 𝑦(𝑎 + 𝑏) + 𝑧(𝑎 + 𝑏)

=(𝑎 + 𝑏)(𝑥 − 𝑦 + 𝑧)

36. 𝑎𝑥 − 𝑏𝑥 + 𝑏𝑦 + 𝑐𝑦 − 𝑐𝑥 − 𝑎𝑦

Refer Que 35

37. (𝑎𝑥 + 𝑏𝑦)2 − (𝑏𝑥 + 𝑎𝑦)2

=[𝑎𝑥 + 𝑏𝑦 + 𝑏𝑥 + 𝑎𝑦][(𝑎𝑥 + 𝑏𝑦) − (𝑏𝑥 − 𝑎𝑦)]

=[𝑥(𝑎 + 𝑏) + 𝑦(𝑎 + 𝑏)][𝑥(𝑎 − 𝑏) − 𝑦(𝑎 − 𝑏)]

=(𝑎 + 𝑏)(𝑥 + 𝑦)(𝑎 − 𝑏)(𝑎 − 𝑦)

Factorization Using Identifiers

Factorization when the given expression is a perfect square.

We know that

A] 𝑎2 + 2𝑎𝑏 + 𝑏2 = (𝑎 + 𝑏)2

B] 𝑎2 − 2𝑎𝑏 + 𝑏2 = (𝑎 − 𝑏)2

Exercise 15B

Factorise the following algebraic expression

1. 𝑥2 − 16𝑥 + 64

= (𝑥)2 − 2 × 8 × 𝑥 + (8)2

= (𝑥 − 8)2



Grade: VIII

2. 1 − 2𝑥 + 𝑥2

= (1)2 − 2 × 1 × 𝑥 + (𝑥)2

= (1 − 𝑥)2

3. 9 + 6𝑝 + 𝑝2

Refer que 2

4. 𝑎2 − 10𝑎 + 25

Refer que 1

5. 1 − 6𝑥 + 9𝑥2

= (1)2 − 2 × 3 × 𝑥 + (3𝑥)2

= (1 − 3𝑥)2

6. 49𝑥2 − 14𝑥 + 1

= (7𝑥)2 − 2 × 7 × 𝑥 + (1)2

= (7𝑥 − 1)2

7. 49𝑥2 − 84𝑥𝑦 + 36𝑦2

= (7𝑥)2 − 2 × 7𝑥 × 6𝑦 + (6𝑦)2

= (7𝑥 + 6𝑦)2

8. 4𝑦2 − 8𝑦 + 4

Refer que 6

9. (𝑎 + 𝑏)2 − 4𝑎𝑏

= 𝑎2 − 2𝑎𝑏 + 𝑏2 − 4𝑎𝑏

= 𝑎2 − 2𝑎𝑏 + 𝑏2

= (𝑎 − 𝑏)2

10. 𝑥4 − 2𝑥2𝑦2 + 𝑦4

= ((𝑥)2)2 + 2 × 𝑥2 × 𝑦2 + ((𝑦)2)2

= (𝑥2 + 𝑦2) 2



Grade: VIII

11. 𝑥2

4+ 2𝑥2𝑦2 + 𝑦4

= (𝑥2)2 + 2 × 𝑥2 × (𝑦2) 2

= (𝑥

2+ 1) 2

12. 𝑥2𝑦2 − 6𝑥𝑦𝑧 + 𝑦2

= (𝑥𝑦)2 − 2 × 𝑥𝑦 × 3𝑧 + (3𝑧)2

= (𝑥𝑦 − 3𝑧) 2

13. 𝑎2

4− 3𝑎𝑏3 + 9𝑏6

= (𝑎

2) 2 − 2 ×

𝑎

2× 3𝑏3 + (3𝑏3) 2

= (𝑎

2− 3𝑏3) 2

14. (𝑎2

9) + (

𝑏2

64) + (

𝑎𝑏

12)

=𝑎2

9+

𝑎𝑏

12+

𝑏2

64

= (𝑎

3) 2 + 2 ×

𝑎

3×

𝑏

8+ (

𝑏

8) 2

= (𝑎

3−

𝑏

8) 2

15. 𝑎2

16−

𝑎𝑏

6+

𝑏2

9

Refer que 1

* When the given expression is a difference of two squares:

In this case we use identity

𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

Exercise 15C

Factorise the following Algebraic Expressions:-

1. 4𝑥2 − 9𝑦2

= (2𝑥)2 − (3𝑦)2

= (2𝑥 + 3𝑦)(2𝑥 − 3𝑦)

2. 81𝑝2 − 121

= (9𝑝)2 − (11)2



Grade: VIII

3. 100 − 49𝑎2

= (10)2 − (7𝑎)2

= (10 − 7𝑎)(10 + 7𝑎)

4. 𝑎5 − 16𝑎3

= 𝑎3[𝑎2 − 16]

= 𝑎3[𝑎2 − 42]

= 𝑎3(𝑎 − 4)(𝑎 + 4)

5. 12𝑥2 − 75𝑦2

= (√12𝑥)2

− (√75𝑦)2

= (√4 × 3𝑥)2

− (√25 × 3𝑦)2

= (2√3𝑥) 2 − (5√3𝑦) 2

= [(2√3𝑥) − (5√3𝑦) ] [ (2√3𝑥) + (5√3𝑦)]

6. 𝑥3 − 64𝑥

Refer que 4

7. 15𝑥5 − 135𝑥3

Refer que 4

8. 81𝑝7 − 𝑝3𝑞4

= 𝑝3(81𝑝4 − 𝑞4)

= 𝑝3 [ (9𝑝)2 − (𝑞2) 2 ]

= 𝑝3 (9𝑝 − 𝑞2)(9𝑝 + 𝑞2)

9. 𝑥2

36−

𝑦2

25

= (𝑥

6)2 − (

𝑦

5)2

= (𝑥

6+

𝑦

5) (

𝑥

6−

𝑦

5)

10. 49x2

144−

16x2

25

Refer que. 9



Grade: VIII

11. 𝑥2

8−

𝑦2

32

Refer que. 9

12. 1 − (𝑥 − 𝑦)2

= (1)2 − (𝑥 − 𝑦)2

= (1 + (𝑥 − 𝑦))(1 − (𝑥 − 𝑦))

= (1 + 𝑥 − 𝑦)(1 − 𝑥 + 𝑦)

13. (2𝑥 + 3𝑦)2 − 16𝑧2

Refer que. 12

14. 4(𝑥 − 𝑦)2 − (𝑥 + 𝑦)2

= (2(𝑥 − 𝑦))2 − (𝑥 + 𝑦)2

= (2𝑥 − 2𝑦 + 𝑥 + 𝑦)(2𝑥 − 2𝑦 − 𝑥 − 𝑦)

= (3𝑥 − 𝑦)(𝑥 − 3𝑦)

15. 25 − 𝑎2 − 2𝑎𝑏 − 𝑏2

= 25 − (𝑎2 + 2𝑎𝑏 + 𝑏2)

= 52 − (𝑎 + 𝑏)2

= (5 + (𝑎 + 𝑏))(5 − (𝑎 + 𝑏))

= (5 + 𝑎 + 𝑏)(5 − 𝑎 − 𝑏)

16. 𝑥4 + 𝑦4 + 𝑥2𝑦2

Add and subtract 2𝑥2𝑦2

= (𝑥)22+ 2𝑥2𝑦2 + (𝑦)22

+ 𝑥2𝑦2 − 2𝑥2𝑦2

= (𝑥2 + 𝑦2)2 − 𝑥2𝑦2

= [(𝑥2 + 𝑦2) + 𝑥𝑦][(𝑥2 + 𝑦2) − 𝑥𝑦]

17. 𝑥4 + 𝑥2 + 1 [𝐻𝑖𝑛𝑡: 𝑎𝑑𝑑 𝑎𝑛𝑑 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑥2]

Refer que 16.

18. 𝑥4 + 4 [𝐻𝑖𝑛𝑡: 𝑎𝑑𝑑 𝑎𝑛𝑑 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 4𝑥2]

Refer que 16.



Grade: VIII

19. Evaluate

(203)2 − (197)2

= (203 + 197)(203 − 197)

= (400)(6)

= 2400

20. Evaluate

(6.6)2 − (3.4)2

Refer que 19.

21. Simplify

0.75 × 0.75 − 0.25 × 0.25

0.75 − 0.25

=(0.75)2 − (0.25)2

0.75 − 0.25

=(0.75 + 0.25)(0.75 − 0.25)

(0.75 − 0.25)

= 0.75 + 0.25

= 1

* Factorization of Quadratic Trinomial

In order to factories quadratic trinomial 𝒙𝟐 + 𝒑𝒙 + 𝒒 , two numbers a&b have to be found such that a+b=p and ab=q

Then, 𝒙𝟐 + 𝒑𝒙 + 𝒒 = 𝒙𝟐 + (𝒂 + 𝒃)𝒙 + 𝒂𝒃[𝒔𝒖𝒃𝒔𝒕𝒊𝒖𝒕𝒊𝒏𝒈 𝒑 & 𝒒]

= (𝒙 + 𝒂)(𝒙 + 𝒃)

Exercise 15D

Factorise the following algebraic expression

1. 𝑥2 + 7𝑥 + 12

= 𝑥2 + 4𝑥 + 3𝑥 + 12

= 𝑥(𝑥 + 4) + 3(𝑥 + 4)

= (𝑥 + 4)(𝑥 + 3)



Grade: VIII

2. 𝑥2 + 9𝑥 + 20

= 𝑥2 + 5𝑥 + 4𝑥 + 20

= 𝑥(𝑥 + 5) + 4(𝑥 + 5)

= (𝑥 + 5)(𝑥 + 4)

3. 𝑝2 − 2𝑝 − 15

= 𝑝2 − 5𝑝 + 3𝑝 − 15

= 𝑝(𝑝 − 5) + 3(𝑝 − 5)

= 𝑝(𝑝 − 5) + 3(𝑝 − 5)

= (𝑝 − 5)(𝑝 + 3)

4. 𝑥2 − 18𝑥 + 65

= 𝑥2 − 13𝑥 − 5𝑥 + 65

= 𝑥(𝑥 − 13) − 5(𝑥 − 13)

= (𝑥 − 13)(𝑥 − 5)

5. 𝑎2 − 11𝑎 − 80

Refer que 3

6. 𝑝2 − 11𝑝 − 102

= 𝑝2 − 17𝑝 + 6𝑝 − 102

= 𝑝(𝑝 − 17) + 6(𝑝 − 17)

= (𝑝 − 17) + 6(𝑝 − 17)

= (𝑝 − 17)(𝑝 + 6)

7. 𝑚2 − 26𝑚 + 120

Refer que 4

8. 3𝑥2 + 10𝑥 + 8

= 3𝑥2 + 6𝑥 + 4𝑥 + 8

= 3𝑥(𝑥 + 2) + 4(𝑥 + 2)

= (𝑥 + 2)(3𝑥 + 4)

9. 3𝑥2 + 10𝑥 + 8

Refer que 8

10. 35 − 33𝑥 + 4𝑥2

= 4𝑥2 − 33𝑥 + 35

= 4𝑥2 − 28𝑥 − 5𝑥 + 35

= 4𝑥(𝑥 − 7) − 5(𝑥 − 7)

= (4𝑥 − 5)(𝑥 − 7)



Grade: VIII

11. 12𝑥2 − 29𝑥 + 15

Refer que 10

12. 4𝑥2 − 8𝑥 + 3

Refer que 10

13. 𝑥2 + 10𝑥𝑦 + 21𝑦2

= 𝑥2 + 7𝑦𝑥 + 3𝑥𝑦 + 21𝑦2

= 𝑥(𝑥 + 7𝑦) + 3𝑦(𝑥 + 7𝑦)

= (𝑥 + 7𝑦)(𝑥 + 3𝑦)

14. 𝑎2 − 𝑎𝑏 − 20𝑏2

= 𝑎2 − 5𝑎𝑏 + 4𝑎𝑏 − 20𝑏2

= 𝑎(𝑎 − 5𝑏) + 4𝑏(𝑎 − 5𝑏)

= (𝑎 − 5𝑏)(𝑎 + 4𝑏)

15. 6 − 2𝑥2 − 𝑥

= −2𝑥2 − 𝑥 + 6

= 2𝑥2 + 4𝑥 − 3𝑥 − 6

= 2𝑥(𝑥 − 2) − 3(𝑥 − 2)

= (𝑥 − 2)(2𝑥 − 3)

16. 14𝑎2 + 11𝑎𝑏 − 15𝑏2

Refer que 10.

17. 36𝑥2 + 12𝑥𝑦𝑧 − 15𝑦2𝑧2

= 3(12𝑥2 + 4𝑥𝑦𝑧 − 5𝑦2𝑧2)

= 3(12𝑥2 + 10𝑥𝑦𝑧 − 6𝑥𝑦𝑧 − 5𝑦2𝑧2)

= 3[2𝑥(6𝑥 + 5𝑦𝑧) − 𝑦𝑧(6𝑥 + 5𝑦𝑧)]

= 3(6𝑥 + 5𝑦𝑧)(2𝑥 − 𝑦𝑧) Division of algebraic Expressions

1. Division of monomial by another monomial:

Quotient of two monomials is the product of the quotient of its numerical coefficients and the quotient of their variables.



Grade: VIII

2. Division of polynomial by a monomial

In order to divide a polynomial by a monomial, each terms of the polynomial is divided by the monomial. 3. Division of a polynomial by a polynomial

In order to divide a polynomial by another polynomial, both the polynomial are factorised and their common factor is cancelled. Division of a polynomial by another polynomial by long division method. The following steps have to be followed to divide a polynomial by another polynomial by long division method. i) Arrange the term of the divisor and the dividend in descending order of their degrees.

ii) In order to obtain the first term of the quotient, divide the first term of the dividend by the first term

of the divisor.

iii) Multiply all the terms of the divisor by the first term of the quotient and subtract the result obtained

from the dividend to get the reminder.

iv) If the reminder is not equal to zero, then consider it a new dividend and proceed as a before.

v) Continue with the same process still a reminder equal to zero or a polynomial of degree less than the

divisor is obtained.

Exercise 15E

1. Divide the following algebraic expressions

i) 𝟑𝟓𝒙𝟔𝒚𝟒 𝑏𝑦 7𝒙𝟐𝒚𝟐

=35𝑥6𝒚𝟒

7𝒙𝟐𝒚𝟐

= (35

7) (

𝑥6

𝑥2) (𝑦4

𝑦2)

= 5𝑥6−2𝑦4−2 = 5𝑥4𝑦2

ii) 12𝑎6𝑏6𝑐5 𝑏𝑦 − 3𝑎4𝑏2𝑐

=12𝑎6𝑏6𝑐5

−3𝑎4𝑏2𝑐

= (12

−3) (

𝑎6

𝑎4)(

𝑏6

𝑏2)(

𝑐5

𝑐)

= (−4)(𝑎6−4)(𝑏6−2)(𝑐5−1)

= −4𝑎2𝑏4𝑐4



Grade: VIII

iii) −63𝑎7𝑏8𝑐3 𝑏𝑦 −9𝑎5𝑏5𝑐3

=−63𝑎7𝑏8𝑐3

−9𝑎5𝑏5𝑐3

= (−63

−9)(

𝑎7

𝑎5)(

𝑏8

𝑏5)(

𝑐3

𝑐3)

= 7𝑎2𝑏3

iv) 16𝑏2𝑦𝑥2𝑏𝑦 − 2𝑥𝑦 Refer que no. iii

v) 4(𝑥4𝑦4 − 2𝑥3𝑦2 + 3𝑥𝑦2)𝑏𝑦 − 2𝑥𝑦 Refer que no. iii

vi) −3𝑥2 + 9

2𝑥𝑦 − 6𝑥𝑧 𝑏𝑦

−3

2𝑥

=−3𝑥2 +

92 𝑥𝑦 − 6𝑥𝑧

−32 𝑥

=−3𝑥2

−32 𝑥

+

92 𝑥𝑦

−32 𝑥

−6𝑥𝑧

−32 𝑥

= −3(2

−3) (

𝑥2

𝑥) +

9

2 (

−2

3) (

𝑥𝑦

𝑥) − 6(

2

−3)(

𝑥𝑧

𝑥)

= 2𝑥 − 3𝑦 + 4𝑧

2. Divide the following

i) (7𝑥 − 35)𝑏𝑦7

=7𝑥 − 35

7

=7𝑥

7−

35

7

= 𝑥 − 5



Grade: VIII

ii) (6𝑥 + 33) ÷ (2x + 11)

=(6𝑥 + 33)

(2x + 11)

=3(2𝑥 + 11)

(2x + 11)

= 3

iii) 72𝑎𝑏(𝑎+5) (𝑏−4)÷ 36𝑎(𝑏 − 4)

=72ab(a + 5)(b − 4)

36𝑎(𝑏 − 4)

= 2𝑏(𝑎 + 5)

iv) 28(𝑥 + 3)(𝑥2 + 3𝑥 + 7) ÷ 7(𝑥 + 3)

=28(𝑥 + 3)(𝑥2 + 3𝑥 + 7)

7(𝑥 + 3)

= 28(𝑥2 + 3𝑥 + 7)

3. Factorise the expression and divide as directed.

i) (𝑥2 − 23𝑥 + 132) ÷ (𝑥 − 11)

Let’s factorise (𝑥2 − 23𝑥 + 132) (𝑥2 − 23𝑥 + 132) = (𝑥2 − 12𝑥 − 11𝑥 + 132)

= 𝑥(𝑥 − 12) − 11(𝑥 − 12) = (𝑥 − 12)(𝑥 − 11)

(𝑥2 − 23𝑥 + 132) ÷ (𝑥 − 11)

=(𝑥2 − 23𝑥 + 132)

(𝑥 − 11)

=(𝑥 − 12)(𝑥 − 11)

(𝑥 − 11)

= (𝑥 − 12)

ii) 3𝑦2 + 12𝑦 − 63 ÷ (𝑦 + 7)

Refer que no. i



Grade: VIII

iii) 14𝑦𝑧(𝑧2 − 5𝑧 − 36) ÷ 7𝑧(𝑧 − 9)

Let’s factorise (𝑧2 − 5𝑧 − 36) (𝑧2 − 5𝑧 − 36) = (𝑧2 − 9𝑧 + 4𝑧 − 36) = 𝑧(𝑧 − 9) + 4(𝑧 − 9) = (𝑧 − 9)(𝑧 + 4)

14𝑦𝑧(𝑧2 − 5𝑧 − 36) ÷ 7𝑧(𝑧 − 9)

=14𝑦𝑧(𝑧2 − 5𝑧 − 36)

7𝑧(𝑧 − 9)

=14𝑦𝑧(𝑧 − 9)(𝑧 + 4)

7𝑧(𝑧 − 9)

= 2𝑦(𝑧 + 4)

iv) 15𝑎𝑏(25𝑎2 − 16𝑏2) ÷ 5𝑎𝑏(5𝑎 + 4𝑏)

Refer que. No. iii

4. Divide the following polynomials by binomials.

i) 𝑥2 − 11𝑥 + 30 𝑏𝑦 (𝑥 − 5)

Let’s factorise 𝑥2 − 11𝑥 + 30 = 𝑥2 − 6𝑥 − 5𝑥 + 30 = (𝑥 − 6)(𝑥 − 5)

𝑥2 − 11𝑥 + 30 𝑏𝑦 (𝑥 − 5)

=𝑥2−11𝑥+30

(𝑥−5)

=(𝑥−6)(𝑥−5)

(𝑥−5)

= (𝑥 − 6)

ii) 2𝑦2 + 17𝑦 + 21 𝑏𝑦 2𝑦 + 3

Let’s factorise 2𝑦2 + 17𝑦 + 21 = 2𝑦2 + 14𝑦 + 3𝑦 + 21

= 2𝑦(𝑦 + 7) + 3(𝑦 + 7) = (𝑦 + 7)(2𝑦 + 3)



Grade: VIII

2𝑦2 + 17𝑦 + 21 𝑏𝑦 2𝑦 + 3

=2𝑦2 + 17𝑦 + 21

2𝑦 + 3

=(𝑦 + 7)(2𝑦 + 3)

2𝑦 + 3

= 𝑦 + 7

iii) 6𝑎2 − 7𝑎 − 3 𝑏𝑦 2𝑎 − 3

Let’s factorise 6𝑎2 − 7𝑎 − 3 = 6𝑎2 − 9𝑎 + 2𝑎 − 3 = 3𝑎(2𝑎 − 3) + (2𝑎 − 3) = (2𝑎 − 3)(3𝑎 + 1)

6𝑎2 − 7𝑎 − 3 𝑏𝑦 2𝑎 − 3

=6𝑎2 − 7𝑎 − 3

2𝑎 − 3

=(2𝑎 − 3)(3𝑎 + 1)

2𝑎 − 3

= 3𝑎 + 1

iv) 15 + 3𝑥 − 7𝑥2 − 4𝑥3 𝑏𝑦 (5 − 4𝑥)

𝑥2 + 3𝑥 + 3 −4𝑥 + 5 −4𝑥3 − 7𝑥2 + 3𝑥 + 15 −4𝑥3 + 5𝑥2 + -

-12𝑥2 + 3𝑥 + 15 -12𝑥2 + 15𝑥 + - −12𝑥 + 15 −12𝑥 + 15 + - 0 0



Grade: VIII

5. Divide the following polynomials by trinomials.

i) 𝑦2 − 6𝑦2 + 11𝑦 − 6 𝑏𝑦 𝑦2 − 4𝑦 + 3

𝑦 − 2 𝑦2 − 4𝑦 + 3

𝑦3 − 6𝑦2 + 11𝑦 − 6 𝑦3 − 4𝑦2 + 3

_ + _ -2𝑦2 + 11𝑦 − 9 -2𝑦2 + 8𝑦 − 6 + _ + 3𝑦 − 3

ii) 𝑥3 − 14𝑥2 + 37𝑥 − 26 𝑏𝑦 𝑥2 − 12𝑥 + 13 Refer que no. i

6. The product of two expressions is𝑦2 − 𝑧2 + 2𝑥𝑦 − 2𝑥𝑧. If one of them is 2𝑥 + 𝑦 + 𝑧 find the other.

𝑦 − 𝑧 2𝑥 + 𝑦 + 𝑧 2𝑥𝑦 − 2𝑥𝑧 + 𝑦2 − 𝑧2 2𝑥𝑦 + 𝑦2 + 𝑦𝑧

- - -

−2𝑥𝑧 −𝑧2 − 𝑦𝑧 −2𝑥𝑧 −𝑧2 − 𝑦𝑧 + + +

0 0 0

7. Find the value of a so that 1 − 7𝑥 is a factor −14𝑥3 − 47𝑥2 − 14𝑥 + 𝑎

Refer que 6.

LESSON NOTES 2021 2022 Subject: Mathematics Chapter: 15 ...

Documents

Transcript of LESSON NOTES 2021 2022 Subject: Mathematics Chapter: 15 ...