# Rd Sharma Solutions Class 8 Chapter 7 Factorization

Welcome to NCTB Solution. Here with this post we are going to help 8th class students for the Solutions of RD Sharma Class 8 Mathematics, Chapter 7, Factorization. Here students can easily find Exercise wise solution for chapter 7, Factorization. Students will find proper solutions for Exercise 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8 and 7.9 Our teacher’s solved every problem with easily understandable methods so that every students can understand easily.

Factorization Exercise 7.1 Solution :

Question no – (1)

Solution :

Given, 2x² and 12x²

= 2x² = 2 × x × x

12x² = 3 × 2 × 2 × x × x

HCF = 2 × x × x = 2x²

Therefore, the greatest common factor of 2x² and 12x² is 2x².

Question no – (2)

Solution :

Given, 6x³y and 18x²y³

= 6x³y = 2 × 3 × x × x × x × y

= 18x²y³ = 2 × 2 × 3 × x × x × y × y × y

HCF = 2 × 3 × x × x × y = 6x²y

Therefore, Greatest common factor of 6x³y and 18x² y³ is 6x²y

Question no – (5)

Solution :

Given, 12ax², 6a²x³ and 2a³x⁵

12ax² = 3 × 2 × a × x × x

6a²x³ = 3 × 2 × a × a × x × x × x

2a³x⁵ = 2 × a × a × a × x × x × x × x × x

HCF = 2 × a × x × x = 2ax²

Therefore, the greatest common factor of 12ax², 6a²x³ and 2a³x⁵ is 2ax².

Question no – (6)

Solution :

Given, 9x², 15x²y³, 6xy² and 21x²y²

9x² = 3 × 3 × x × x

15x²y³ = 3 × 5 × x × x × y × y × y

6xy² = 3 × 2 × x × y × y × y

21x²y² = 3 × 7 × x × x × y × y

HCF = 3 × x = 3x

Therefore, the greatest common factor of 9x², 15x²y³, 6xy² and 21x²y² is 3x.

Question no – (7)

Solution :

As per the question, 4a²b³, -12a³b, 18a⁴b³

4a²b³ = 2 × 2 × a × a × b × b × b

12a³b = 3 × 2 × 2 × a × a × a × b

18a⁴b³ = 3 × 3 × 2 × a ×a × a × a × b × b × b

HCF = 2 × a × a × b = 2a²b

Therefore, Greatest Common Factor of 4a²b³, -12a³b and 18a⁴b³ is 2a²b

Question no – (8)

Solution :

Given, 6x²y², 9xy³, 3x³y²

6x²y² = 2 × 3 × x × x × y × y

9xy³ = 3 × 3 × x × y × y × y

3x³y² = 3 × x × x × x × y × y

HCF = 3 × x × y × y × = 3xy²

Therefore, Greatest common factor of 6x²y², 9xy³ and 3x³y² is 3xy²

Question no – (9)

Solution :

Given, a²b³ and a³b²

a²b³ = a × a × b × b × b

a³b² = a × a × a × b × b

HCF = a × a × b × b = a²b²

Therefore, Greatest Common Factor of a²b³ and a³b² is a²b²

Question no – (10)

Solution :

Given, 36a²b²c⁴, 54a⁵c², 90a⁴b²c²

36a²b²c⁴ = 2 × 2 × 3 × 3 × a × a × b × b × c × c × c × c

54a⁵c² = 2 × 3 × 3 × 3 × a × a × a × a × a × c × c

90a⁴b²c² = 2 × 3 × 3 × 5 × a × a × a × a × b × b × c × c

HCF = 2 × 3 × 3 × a × a × c × c = 18a²c²

The greatest common factor of 36a²b²c⁴, 54a⁵c² and 90a⁴b²c² is 18a²c²

Question no – (11)

Solution :

Given, x³ and -yx²

x³ = x × x × x

-yx² = y × x × x

HCF = x × x = x²

The greatest common factor of x³ and – yx² is x²

Question no – (12)

Solution :

Given, 15a³, -54a², -150a

15a³ = 3 × 5 × a × a × a

-45a² = 3 × 3 × 5 × a × a

– 150a = 3 × 5 × 2 × 5 × a

HCF = 3 × 5 × a = 15a

Therefore, the Greatest Common Factor of 15a³, -45a² and -150a is 15a

Question no – (13)

Solution :

Given, 2x³y², 10x²y³, 14xy

2x³y² = 2 × x × x × x × y × y

10x²y³ = 2 × 5 × x × x × y × y × y

14xy = 2 × 7 × x × y

HCF = 2 × x × y = 2xy

Therefore, Greatest Common Factor of 2x³y², 10x²y³ and 14xy is 2xy.

Question no – (14)

Solution :

Given, 14x³y⁵, 10x⁵ y³, 2x²y²

14x³y⁵  = 2 × 7 × x × x × x × y × y × y × y× y

10x⁵y³ = 2 × 5 × x × x × x × x × x × y × y × y

2x²y² = 2 × x × x × y× y

HCF = 2 × x × x × y × y = 2x²y²

Therefore, the Greatest Common Factor of 14x³y⁵, 10x⁵ y³ and 2x²y² is 2x²y²

Question no – (15)

Solution :

5a⁴ + 10a³ – 15a²

= 5 × a × a × a × a + 2 × 5 × a × a × a – 3 × 5 × a × a

= 5a² (a² + 2a – 3)

The Greatest Common Factor will be 5a²

Question no – (16)

Solution :

Given, 2xyz + 3x²y + 4y²

= 2 × x × y × z + 3 × x × x × y + 2 × 2 × y × y

= y (2xz + 3x² + 4y)

The Greatest Common Factor is y

Question no – (17)

Solution :

Given, 3a²b² + 4b²c² + 12a²b²c²

= 3 × a × a × b × b + 4 × b × b × c × c + 3 × 4 × a × a × b × b × c × c

= b² (3a²+ 4c² + 12a²c²)

The Greatest Common Factor of the given number b²

Factorization Exercise 7.2 Solution :

Question no – (1)

Solution :

Given, 3x – 9

= 3x – 3²

3 (x – 3)

Question no – (2)

Solution :

Given, 5x – 15x²

5x (1 – 3x)

Question no – (3)

Solution :

Given, 20a^12b² – 15a⁸b⁴

5a³b² (4a⁴- 3b²)

Question no – (4)

Solution :

Given, 72x⁶y⁷ – 96x⁷y⁶

24x⁶y⁶ (3y – 4x)

Question no – (5)

Solution :

Given, 20x³ – 40² + 80x

20x (x² – 2² + 4)

Question no – (6)

Solution :

Given, 2x³y² – 4x²y³ + 8xy⁴

2xy² (x – 2xy + 4y²)

Question no – (7)

Solution :

Given, 10m³n² +15m⁴n – 20m²n³

5m²n (2mn+3m² – 4n²)

Question no – (8)

Solution :

Given, 2a⁴b⁴ – 3a³b⁵ – 4a²b⁵

a²b⁴ (2a² – 3ab + 4b)

Question no – (9)

Solution :

Given, 28a² + 14a²b² – 21a⁴

7a² (a + 2b² – 3a²)

Question no – (10)

Solution :

Given, a⁴ b – 3a²b² – 6ab³

ab (a³ – 3ab – 6b²)

Question no – (11)

Solution :

Given, 2l²mn – 3lm²n + 4lmn²

lmn (2l – 3m + 4n)

Question no – (12)

Solution :

Given, x⁴y² – x²y⁴ – x⁴y⁴

∴ x²y² (x² – y² – x²y²)

Question no – (13)

Solution :

Given, 9x²y + 3axy

∴ 3xy (3x + a)

Question no – (14)

Solution :

Given, 16m – 4m²

4m (4 – m)

Question no – (15)

Solution :

Given, – 4a² – 4ab – 4ca

– 4a (a – b + c)

Question no – (16)

Solution :

Given, x²yz + xy²z + xyz²

xyz (x+ y+ z)

Question no – (17)

Solution :

Given, ax²y + bxy² + cxyz

xy (ax + by + cz)

Factorization Exercise 7.3 Solution :

Question no – (1)

Solution :

Give, 6x (2x-y) + 7y (2x-y)

(2x – b) (6x + 7y)

Question no – (2)

Solution :

Given, 2r(y – x) + s (x – y)

= 2r(b – x) – s (y – x)

(y – x) (2r – s)

Question no – (3)

Solution :

Given, 7a(2x – 3) + 3b (2x – 3)

(2x + 3) (7a + 3b)

Question no – (4)

Solution :

Given, – 9a (6a – 5b) – 12a²(6a – 5b)

= (6a – 5b) (9a – 12a²)

3a(3 – 4a) (6a – 5b)

Question no – (5)

Solution :

Given, 5(x – 2y)² + 3 (x – 2y)

= (x – 2y) [(x – 2y) 5 + 3]

∴ (x – 2y) (5x – 10y + 3)

Question no – (6)

Solution :

Given, 16(2l – 3m) ² – 12(3m – 2l)

= 16 (2l – 3m) ² + 12 (2l – 3m)

= 4 (2l – 3m) [4 (2l – 3m) + 3]

4 (2l – 3m) (8l – 12m + 3)

Question no – (7)

Solution :

Given, 3a (x – 2y) – b (x – 2y)

(x – 2y) (3a – b)

Question no – (8)

Solution :

Given, a²(x + y) + b²(x + y) + c²(x + y)

(x + y) (a² + b² + c²)

Question no – (9)

Solution :

Given, (x – y)² + (x – y)

(x – y) (x – y + 1)

Question no – (10)

Solution :

Given, 6 (a + 2b) – 4 (a² + 2b) ²

= 2 (a + 2b) (3 – 2 (a + 2b))

2 (a + 2b) (3 – 2a – 4b)

Question no – (11)

Solution :

Given, a (x – y) + 2b(y – x) + c(x – y)²

= a (x – y) – 2b (x – y) + c (x – y)²

= (x – y) [a – 2b + c (x – y)]

(x – y) (a – 2b) + (cx – cy)

Question no – (12)

Solution :

Given, – 4 (x – 2y)² + 8(x – 2y)

– 4 (x – 2y) (x – 2y – 2)

Question no – (13)

Solution :

Given, x³(a – 2b) + x²(a – 2b)

x²(a – 2b) (x + 1)

Question no – (14)

Solution :

Given, (2x – 3y) (a + b) + (3x – 2y) (a + b)

= (a + b) (2x – 3y + 3x – 2y)

= (a + b) (5x – 5y)

5 (x + y) (a + b)

Question no – (15)

Solution :

Given, 4(x + y) (3a – b) + 6 (x + y) (2b – 3a)

= 2(x + y) [2 (3a – b) + 3 (2b – 3a)]

= 2(x + y) (6a – 2b + 6b – 9a)

2(x + y) (4b – 3a)

Factorization Exercise 7.4 Solution :

Question no – (1)

Solution :

Given, qr – pr + qs – ps

= r(q – p) + s(q – p)

(q – p) (r + s)

Question no – (2)

Solution :

Given, p²q – pr² – pq + r²

= p (q p – r²) – 1 (p q- r²)

(pq – r²) (p – 1)

Question no – (3)

Solution :

Given, 1 + x + xy + x²y

= xy (x + 1) + 1(x + 1)

(x + 1) (xy + 1)

Question no – (4)

Solution :

Given, ax + ay – b x – by

= a(x + y) – b(x + y)

(x + y) (a – b)

Question no – (6)

Solution :

Given, x² + xy + xz + yz

= x(x+ y) + z(x + y)

(x + y) (x + z)

Question no –  (7)

Solution :

Given, 2ax + bx + 2ay + by

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

(2a + b) (x + y)

Question no – (8)

Solution :

Given, ab – by – ay + y²

= b(a – y) – y(a – y)

(a – y) (b – y)

Question no – (9)

Solution :

Given, axy + bcxy – az – bcz

= axy – az + bcxy – bcz

= a(xy – z) + bc(xy – z)

(xy – z) (a + bc)

Question no – (10)

Solution :

Given, 2l²mn – 3lm²n + 4lmn²

lmn (2l – 3m + 4n)

Question no – (11)

Solution :

Given, x³ – y² +x – x²y²

= x³ + x – y² – x²y²

= x(x² + 1) – y²(x² + 1)

(x² + 1) (x – y²)

Question no – (12)

Solution :

Given, 6xy + 6 – 9y – 4x

= 6xy – 4x + 6 – 9

= 2x (3y – 2) – 3 (3y – 2)

(3y – 2) (2x – 3)

Question no – (13)

Solution :

Given, x² – 2ax – 2ab + bx

= x (x – 2a) – b(x – 2a)

(x – 2a) (x – b)

Question no – (14)

Solution :

Given, x³ – 2x²y + 3xy² – 6y³

= x³ + 3xy² – 2x²y – 6y³

= x (x² + 3y) – 2y (x² + 3y²)

∴ (x – 2y) (x² + 3y²)

Question no – (15)

Solution :

Given, abx² + (ay – b) x – y

= abx² + ayx – bx – y

= ax (bx+ y) -1 (bx + y)

(bx + y) (ax – 1)

Question no – (16)

Solution :

Given, a²x² + 2abxy + b²y² + b²y² – 2abxy + a²y²

= a²x² + b²x² + b²y + a²y²

= x² (a² + b²) + y² (a² + b²)

(x² + y²) (a² + b²)

Question no – (17)

Solution :

Given, 16 (a – b)³ – 24 (a – b)²

= 8 (a – b)² [2 (a – b) – 3]

8 (a – b)² (2a – 2b – 3)

Question no – (18)

Solution :

Given, ab (x² + 1) + x(a² + b²)

= abx² + a b +a²x + b²x

= abx² + a²x + ab + b²

= ax(bx + a) + b (bx + a)

(bx + a) (ax + b)

Question no – (19)

Solution :

Given, a²x² + (ax² + 1) x + a

= a²x² + ax³ + x + a

= ax² (a + x) + 1 (a + x)

(a + x) (ax² + 1)

Question no – (20)

Solution :

Given, a (a – 2b – c) 2bc

= a² – 2ab – ac + 2bc

= a (a – 2b) – c (a – 2b)

(a – 2b) (a – c)

Question no – (21)

Solution :

a (a + b – c) – bc

= a² +ab – ac – bc

= a(a+ b) – c (a + b)

(a + b) (a – c)

Question no – (22)

Solution :

Given, x² – 11xy – x + 11y

= x(x – 11y) – 1(x – 11y)

(x – 11y) (x – 1)

Question no – (23)

Solution :

Given, ab – a – b +1

= a (b – 1) – 1 (b – 1)

(b – 1) (a – 1)

Question no – (24)

Solution :

Given, 1 + x + xy + x²y

= xy (x + 1) + 1(x + 1)

(x + 1) (xy + 1)

Factorization Exercise 7.5 Solution :

Question no – (1)

Solution :

Given, 16x² – 25y²

= (4x)² – (5y)²

(4x + 5y) (4x – 5y)

Question no – (2)

Solution :

Given, 27x² – 12y²

= 3(9x² – 4y²)

= 3[(3x) ² – (2y) ² ]

3[(3x + 2y) (2x – 2y)

Question no – (3)

Solution :

Given, 144a² – 289b²

= (12a)² – (17b)²

∴ 12a + 17b) (12a – 17b)

Question no – (4)

Solution :

Given, 12m² – 27

= 3 (4m² – 9)

= 3 [ (2m)² – 3²]

3 (2m + 9) (2m – 9)

Question no – (5)

Solution :

Given, 125x² – 45y²

= 5 (25x² – 9y²)

= 5 [(5x)² – (3y)²]

5 (5x + 3y) (5x – 3y)

Question no – (6)

Solution :

Given, 144a² – 169b²

= (12a)² – (13b)²

(12a + 13b) (12a – 13b)

Question no – (7)

Solution :

Given, (2a – b)² – 16c

= (2a -b)² – (4c)²

(2a – b + 4c) (2a – b – 4c)

Question no – (8)

Solution :

Given, (x + 2y)² – 4 (2x – y)²

= x² + 2xy + 4y² – 4(4x² – 2xy +y²)

x² + 2xy + 4y² – 16x²

Question no – (9)

Solution :

Given, 3a⁵ – 48a³

= 3a³ (a² – 16)

= a² – 4²

(a + 4 ) (a – 4)

Question no – (10)

Solution :

Given, a⁴ – 16b⁴

= (a²)² – (4v²)² = (a² + 4b²)

= (a² + 4b²) [a² – (2b) ² ]

(a² + 4b²) (a² + 2b) (a² – 2b)

Question no – (11)

Solution :

Given, x⁸ – 1

= (x⁴)² – 1² = (x⁴ + 1) (x⁴ – 1)

= (x⁴ +1) [(x⁴)⁴ – 1⁴]

= (x⁴ +1) (x² +1) (x² – 1)

= (x⁴ +1) (x² +1) (x² -1)

(x² +1) (x² + 1) (x + 1) (x – 1)

Question no – (12)

Solution :

Given, 64 – (a + 1)²

= 8² – (a + 1)²

= (8 + a + 1) (8 – a – 1)

(a + 9) (7 – a)

Question no – (13)

Solution :

Given, 36l² 36l² – (m+ n)²

= (6l) ² – (m +n)²

= (6l + m + n) (6l – m -n) (m+ n)²

= (6l) ² – (m +n)²

(6l + m + n) (6l – m -n)

Question no – (14)

Solution :

Given, 25x⁴ y⁴ – 1

= (5x²y²)² – 1

(5x²y² + 1) (5x²y² – 1)

Question no – (15)

Solution :

Given,  a⁴ – 1/b⁴

= (a²)² – (1/b²)²

= (a² + 1/b²) (a² – 1/b²)

(a² + 1/b²) (a + 1/b) (a – 1/b)

Question no – (16)

Solution :

Given, x³ – 144x

= x (x² – 144)

= x (x² – 12²)

x (x + 12) (x – 12)

Question no – (18)

Solution :

Given, 9 (a – b)² – 100(x -y)²

= [3 (ab)]² – [10 (x – y)]²

[3 (a – b) + 10(x – y) ] [3 (a-b) – 10 (x – y)]

Question no – (19)

Solution :

Given, (3 + 2a)² – 25a²

= (3 + 2a)² – (5a)²

= ( 3 + 2a + 5a) (3 + 2a – 5a)

(3 + 7a) (3 – 3a)

Question no – (20)

Solution :

Given, (x + y)² – (a – b)²

(x+ y + a – b) (x + y – a + b)

Question no – (21)

Solution :

Given, 1/16x²y² – 4/49y²z²

= (4/1xy)² – (2/7yz) ²

(1/4xy + 2/7yz) (1/4xy – 2/7yz)

Question no – (22)

Solution :

Given, 75a³b² – 108ab⁴

= 3ab² (25²a – 36b²)

= 3ab² [(5a)² – (6b)²]

3ab² (5a + 6b) (5a – 6b)

Question no – (23)

Solution :

Given, x⁵ – 16x³

= x³ (x² – 16)

= x³ (x² – 4²)

= x³ (x² + 4) (x² – 4)

= x³ (x² + 4) (x² + 2²)

x³ (x + 2) (x – 2)

Question no – (24)

Solution :

Given, 50/x² – 2x²/81

= 2 (25/x² – x²/81)

= 2 [(5/x)² – (x/9)²]

2 (5/x + x/9) (5/x – x/9)

Question no – (28)

Solution :

Given, p²q² – p⁴q⁴

= (pq)² – (p²q²)

= p²q² (1- p²q²)

= p² q² [(1² – (pq)²]

p²q² (1 + pq) (1 – p q)

Question no – (30)

Solution :

Given, a⁴b⁴ – 16c⁴

= (a²b²) – (4c)²

= (a²b² + 4c²) (a²b²- 4c²)

= (a²b² + 4c²) [(ab)² – (2c)² ]

(a²b² + 4c²) (ab + 2c) (ab – 2c)

Question no – (31)

Solution :

Given, x⁴ – 625

= (x²)² – (25)²

= (x² +25) (x² – 25)

= (x² + 25) (x² – 5²)

(x² + 25) (x + 5) (x – 5)

Question no – (32)

Solution :

Given, x⁴ – 1

= (x²)² – 1²

= (x² + 1) (x² – 1)

= (x² + 1) (x² – 1²)

(x² + 1) (x + 1) (x – 1)

Question no – (34)

Solution :

Given, x – y – x² + y²

= x – y -(x² – y²)

= (x – y) – (x + y) (x – y)

(x – y) (1 – x + y)

Question no – (35)

Solution :

Given, 16 (2x – 1)² – 25y²

= [4(2x – 1)]² – (5y)²

= [4(2x – 1) + 5y)] [4(2x – 1) – 5y]

(8x – 4 + 5y) (8x – 4 – 5y)

Question no – (37)

Solution :

Given, (2x + 1)² – 9x⁴

= (2x + 1)²- (3x²)²

(2x +1 + 3x²) (2x + 1 – 3x²)

Question no – (38)

Solution :

Given, = x⁴ + (2y – 3z)

= (x²) – (2y – 3z)²

(x² + 2y – 3z) (x² – 2y + 3z)

Question no – (39)

Solution :

Given, a² – b² + a – b

= (a² – b²) + (a – b)

= (a + b) (a – b) + (a + b)

(a – b) (a + b + 1)

Question no – (40)

Solution :

Given, 16a⁴ – b⁴

= (4a²) – (b²)

= (4a² + b²) (4a² – b²)

= (4a² + b²) [(2a)² – b²]

(4a² + b²) (2a + b) (2a – b)

Question no – (42)

Solution :

Given, 2a⁵ – 32a

= 2a (2a⁴ – 160

= 2a [a²) – 4²]

= 2a (a² +4) (a² -4)

= 2a (a² +4) (a² – 2²)

2a (a² + 4) (a  + 2) (a – b)

Question no – (43)

Solution :

As per the question, a⁴b⁴ – 81c⁴

= (a²b²)² – (9c²)

= (a² b² + 9c²) (a² b² – 9c²)

= (a² b² +9c²) [(ab) ² – (3c) ²]

(a² b² + 9c²) (ab + 3c) (ab – 3c)

Question no – (44)

Solution :

Given, xy⁹ – yx⁹

= xy (y⁸ – x⁸) = xy [(y⁴)² – (x⁴)²]

= xy (y⁴ + x⁴) (y⁴ – x⁴)

= xy (y⁴ + x⁴) (y⁴ + x⁴) [(y²)² – (x²)²]

= xy (y⁴ + x⁴) (y² + x²) (y² – x²)

xy (y⁴ + x⁴) (y² – x²) (y + x) (y – x)

Question no – (45)

Solution :

Given,  x³ – x

= x (x² – 1)

= x (x² – 1²)

x (x + 1) (x – 1)

Question no – (46)

Solution :

Given, 18a²x² – 32

= 2 (9a²x² – 16)

= 2 [(3ax)² – 4²]

2 [3ax +4] [3ax – 4)]

Factorization Exercise 7.6 Solution :

Question no – (2)

Solution :

Given, 9a² – 24ab + 16b²

= (3a) ² – 2. 3a. 4b + (4b)²

= (3a – 4b)²

(3a – 4b) (3a – 4b)

Question no – (3)

Solution :

Given in the question, p²q² – 6pqr + 9r²

= (pq)² – 2 . p q. 3r + (3r)²

= (pq – 3r)²

(pq – 3r) (pq – 3r)

Question no – (5)

Solution :

Given,  a² + 2ab + b² – 16

= (a + b)² – 4²

(a + b + 4) (a + b – 4)

Question no – (6)

Solution :

Given, 9z² – x² + 4xy – 4y²

= (3z)² – [x² – 2. x 2y + (2y)²]

= (3z)² – (x – 2y) ²

(3z + x – 2y) (3z – x + 2y)

Question no – (8)

Solution :

As per the question, 16 – a⁶ + 4a³ b³ – 4b⁶

= 4² – [(a³)²- 2. a³ 2b² + (2b³)²]

= 4² – (a³ + 2b³)²

(4 + a³ + 2b³) (4 – a³ – 2b³)

Question no – (9)

Solution :

Given, a² – 2ab + b² – c²

= (a – b)²c²

(a – b + c) (a – b – c)

Question no – (10)

Solution :

Given, x² + 2x + 1 – 9y²

= (x + 1)² – (3y)²

(x + 1 + 3y) (x + 1 – 3y)

Question no – (11)

Solution :

Given, in the question, a² + 4ab + 3b²

= a² + 4ab + 4b² – b²

= a² + 2. a. 2b + (2b)² – b²

= (a + 2b)² – b²

= (a + 2b + b) (a + 2b – b)

(a + 3b) (a + b)

Question no – (12)

Solution :

Given, 96 – 4x – x²

= 100 – 4 – 4x – x²

= (10)² – (2² + 2.2.x + x²)

= (10)² – (2 + x) ²

= (10 + 2 + x) (10 – 2 + x)

(12 + x) (8 + x)

Question no – (13)

Solution :

Given, a⁴ + 3a² + 4

= a⁴ + 4a² + 4 – a²

= [(a²)² + 2 . a² . 2 + (2)²] – a²

= (a² + 2) – a²

(a² + 2 + a) (a² + 2 – a)

Question no – (17)

Solution :

Given, 25 – p² – q² – 2pq

= 5² – (p² + 2pq + q²)

= 5² – (p + q) ²

(5 + p + q) (5 – p – q)

Question no – (18)

Solution :

Given, x² + 9y² – 6xy – 25a²

= [x² – 2. x .3y + (3y)²] -(5a)²

= (x – 3y)² – (5a) ²

∴ (x – 3y + 5a) (x – 3y – 5a)

Question no – (19)

Solution :

Given, 49 – a² + 8ab – 16b²

= 7² – (a² – 82. a. 4b + 4b²)

= 7² – (a – 4b)²

(7 + a – 4b) (7 – a + 4b)

Question no – (20)

Solution :

As per the question, a² – 8ab + 16b² – 25c²

= [a² – 2a. a .4b + (4b)²] – (5c)²

= (a – 4b)² – (5c)²

(a – 4b + 5c) (a – 4b – 5c)

Question no – (21)

Solution :

Given, x² – y² + 6y – 9

= x² – (y² – 2.y.3 + 3²)

= x² – (y – 3) ²

(x + y – 3) (x – y + 3)

Question no – (22)

Solution :

Given,  25x² – 10x + 1 – 36y²

= [(5x)² – 2. 5x. 1+ 1²] – (6y)²

= (5x + 1)² – (6y)²

= (5x + 1)² – (6y)² (5x + 16y)

(5x + 1 + 6y) (5x + 1 – 6y)

Question no – (23)

Solution :

Given, a² – b² + 2bc – c²

= a² – (b² – 2bc + c²)

= a² – (b – c)²

(a + b – c) (a – b + c)

Question no – (24)

Solution :

Given, a² + 2ab + b² – c²

= (a + b)² – c²

(a + b + c) (a + b – c)

Question no – (25)

Solution :

Given, in the question, 49 – x² – y² + 2xy

= 7² – (x² – 2xy + y²)

= 7² – (x – y) ²

(7 + x – y) (7 – x + y)

Question no – (26)

Solution :

Given in the question, a² + 4b² – 4ab – 4c²

= [a² – 2. a. 2b + (2b)²] – (2c)²

= (a – 2b)² – (2c)²

(a – 2b + 2c) (a – 2b – 2c)

Question no – (27)

Solution :

Given, x² – y² – 4xz + 4z²

= [x² – z. x. z + (2z)²]

= (x – 2z)² – y²

(x – 2z + y) (x – 2z – y)

Factorization Exercise 7.7 Solution :

Question no – (2)

Solution :

As per the question, 40 + 3x – x²

= 40 + (8 – 5) x – x²

= 40 + 8x – 5x – x²

= 8(x + 5) – x (x + 5)

(x + 5) (8 – x)

Question no – (3)

Solution :

As per given question, a² – 3a – 88

= a² + (11 – 8) a – 88

= a² + 11a – 8a – 88

= a (a + 11) – 8 (a + 11)

(a + 11) (a – 8)

Question no – (4)

Solution :

Given, a² – 14a – 51

= a² – 2. a .7 + 7² – 7² – 51

= (a – 7)² – 49 – 51

= (a – 7)² – 100 = (a – 7)² – (10)²

= (a – 7 + 10) (a – 7 – 10)

(a + 3) (a – 17)

Question no – (5)

Solution :

Given, x² + 14x + 45

= x² + (9 + 5)x + 45

= x² + 9x + 5x + 45

= x (x + 9) + 5(x + 9)

∴ (x + 9) (x + 5)

Question no – (7)

Solution :

x² – 11x – 42

= x² – (14 – 3) x – 42

= x² – 14x + 3x – 42

= x (x – 14) + 3 (x – 14)

(x – 14) (x + 3)

Question no – (8)

Solution :

Given, a²+ 2a – 3

= a² + (3 – 1) a – 3

= a² + 3a – a – 3

= a(a +3) – 1 (a + 3)

(a+ 3) (a – 1)

Question no – (9)

Solution :

Given,  a² + 14a + 48

= a² + (8 + 6)a + 48

= a² + 8a + 6a + 48

= a(a + 8) + 6(a + 8)

(a + 8) (a + 6)

Question no – (10)

Solution :

Given, x² – 4x – 21

= x² – (7 – 3) x – 21

= x² – 7x + 3x – 21

= x(x – 7) + 3 (x – 7)

(x – 7) (x + 3)

Question no – (11)

Solution :

Given, y² + 5y – 36

= y² (9 – 4)y – 36

=- y² + 9y – 4y – 36

= y(y + 9) – 4(y + 9)

(y + 9) (y – 4)

Question no – (12)

Solution :

Given, (a² – 5a)² – 36

= (a² – 3a – 2a + 6) (a² – 6a + a – 6)

= a(a – 3) – 2 (a – 3) a(a – 6) + 1 (a – 6)

(a – 3) (a – 2) (a – 6) (a + 1)

Question no – (13)

Solution :

As per the question, (a + 7) (a – 10) + 16

= a² – 10 a + 7a – 70 + 16

= a² – 3a – 54

= a² – 9a + 6a – 54

= a (a – 9) + 6(a – 9)

(a – 9) (a + 6)

Factorization Exercise 7.8 Solution :

Question no – (1)

Solution :

Given, 2x² – 3x – 2

= 2x² – 4x + x – 2

= 2x(x – 2) + 1(x – 2)

(2x + 1) (x – 2)

Question no – (3)

Solution :

Given, 3x² + 10x + 3

= 3x² + 9x + x + 3

= 3x(x + 3) + 1(x + 3)

(x + 3) (3x + 1)

Question no – (4)

Solution :

Given, 7x – 6 – 2x²

= – 2x² + 7x – 6

= – 2x² + 3x + 4x – 6

= – x(2x – 3) + 2(2x – 3)

(2x – 3) (2 – x)

Question no – (5)

Solution :

7x² – 19x – 6

= 7x² – 21x + 2x – 6

= 7x (x – 3) + 2 (x – 3)

(x – 3) (7x + 2)

Question no – (6)

Solution :

As per the question, 28 – 31x + 5x²

= 28 – 4x + 35x – 5x²

= 4 (7 – x) + 5x (7 – x)

(7x – x) (4 + 5x)

Question no – (7)

Solution :

Given,  3 + 23y – 8y²

= 3 – y + 24y – 8y²

= -8y² + 24y – y + 3

= -8y (y – 3) – 1 (y – 3)

∴ (y – 3) (1 – 8y)

Question no – (8)

Solution :

Given, 11x² – 54x + 63

= 11x² – 33x – 21x + 63

= 11x(x – 3) – 21(x – 3)

(x – 3) (11x – 21)

Question no – (9)

Solution :

7x – 6x² + 20

= -6x² + 7x + 20

= -6x² – 15 + 8x +20

= – 6x² – 8x + 15x + 20

= – 2x(3x + 4) + 5(3x + 4)

(3x + 4) (5 – 2x)

Question no – (10)

Solution :

3x² + 22x + 35

= 3x² + 15x + 7x + 35

= 3x (x + 5) + 7(x + 5)

(x + 5) (3x + 7)

Question no – (11)

Solution :

Given, 12x² – 17xy + 6y²

= 12x² – 9xy – 8xy + 6y²

= 3x (4x – 3y) – 2y (4x – 3y)

(4x – 3y) (3x – 2y)

Question no – (12)

Solution :

Given in the question, 6x² – 5xy – 6y²

= 6x² – 9xy + 4xy – 6y²

= 3x (2x – 3y) + 2y (2x – 3y)

(2x – 3y) (3x + 2y)

Question no – (13)

Solution :

Given, 6x² – 13xy + 2y²

= 6x² – 12xy – x y + 2y²

= 6x(x – 2y) + y(x – 2y)

(x – 2y) (6x + y)

Question no – (14)

Solution :

Given, 14x² – 13xy + 15y²

= 14x² + 21xy – 10xy – 15y²

= 7x(2x + 3y) – 5y(2x+ 3y)

(2x + 3y) (7x – 5y)

Question no – (15)

Solution :

6a² + 17ab – 3b²

= 6a² – 18ab + ab – 3b²

= 6a (a – 3b) + b(a – 3b)

= (a – 3b) (6a + b)

Question no – (16)

Solution :

Given,  36a² + 12abc – 15b²c²

= 36a² – 18abc + 30abc – 15b²c²

= 18a (2a – bc) + 15bc (2a – bc)

= (18a- 15bc) (2a – bc)

3 (6a – 5bc) (2a – bc)

Question no – (17)

Solution :

Given, 15x² – 16xyz – 15y²z²

= 15x² – 25xyz + 9xyz – 15y²z²

= 5x (83x – 5yz) + 3yz (3x – 5yz)

∴ (5x + 3yz) (3x – 5yz)

Question no – (18)

Solution :

(x – 2y) ² – 5 (x – 2y) + 6

Let, (x – 2y) = a

= a² – 5a + 6

= a² – 3a – 2a + 6

= a (a – 3) – 2 (a – 3)

= (a – 3) (a – 2)

= (x – 2y – 3) (x – 2y – 2)

Question no – (18)

Solution :

Given,  (2a – b)² + 2 (2a – b) – 8

Let, (2a – b) = x

x² + 2 x – 8

= a² + 4 – 2x – 8

= x (x + 4) – 2 (x + 4)

= (x + 4) (x – 2)

= (2a – b + 4) (2a – b – 2)

[ x = (2a – b)]

Question no – (19)

Solution :

Given, x² + 5x + 3

= 2x² + 3x + 2x +3

= x(2x +3) + 1(2x + 3)

(x + 1) (2x + 3)

Factorization Exercise 7.9 Solution :

Question no – (1)

Solution :

Given,  p² + 6p + 8

= p² + 4p + 2p + 8

= p(p + 4) + 2(p + 4)

∴ (p + 4) (p + 2)

Question no – (2)

Solution :

Given, q² – 10q + 21

= q² – 7q – 3q + 21

= q(q – 7) – 3(q – 7)

∴ (q – 7) (q – 3)

Question no – (3)

Solution :

Given, 4y² + 12y + 5

= 4 (y² + 3y 5/4)

= 4 [(y² + 2. 1/2. 3y + (1/2) ² + 5/4]

= 4 [( y+ 1/2 )² – 4/4]

= 4 [(y + 1/2) – 1² ]

= 4 [y + 1/2 + 1] [ y + 1/2 -1]

= 4 [2y + 1 + 2/2] [2y + 1 – 2/2]

= 2 [(2y + 3) (2y – 1)

(4y + 6) (4y – 2)

Question no – (4)

Solution :

Given, p² + 6p – 16

= p² + 2. p. 3 + 3v – 3² – 16

= (p + 3)² – 9 – 16

= (p + 3) ² – 25

= (p + 3) ² – 5²

= (p + 3 + 5) (p + 3 – 5)

(p + 8) (p – 2)

Question no – (5)

Solution :

Given, x² + 12x + 20

= x²+ 2. 2. x. 6 + 6² – 6² – 20

= (x + 6)² – 36 – 20

= (x + 6)² – 16

= (x + 6)² – 4²

= (x + 6 + 4) (x + 6 – 4)

(x + 10) (x + 2)

Question no – (6)

Solution :

Given, a² – 14a – 51

= a² – 2. a .7 + 7² – 7² – 51

= (a – 7)² – 49 – 51

= (a – 7)² – 100 = (a – 7)² – (10)²

= (a – 7 + 10) (a – 7 – 10)

(a + 3) (a – 17)

Question no – (7)

Solution :

Given, a²+ 2a – 3

= a² + (3 – 1) a – 3

= a² + 3a – a – 3

= a(a +3) – 1 (a + 3)

(a+ 3) (a – 1)

Question no – (10)

Solution :

Given, z² – 4z – 12

= z² – 2.z.2 + 2² – 2² – 12

= (z – 2)² – 4 – 12

= (z – 2)² – 16

= (z – 2)² – 4²

= (z – 2 + 4) (z – 2 – 4)

(z + 2) (z – 6)

Next Chapter Solution :

Updated: June 13, 2023 — 3:54 pm