Chapter 7: Factorization Exercise – 7.5

Question: 1

Solve:

16x² – 25y²

Solution:

16x² – 25y²

= (4x)² – (5y)²

= (4x – 5y)(4x + 5y)

Question: 2

Solve:

27x² – 12y²

Solution:

27x² – 12y²

= 3(9x² – 4y²)

= 3[(3x)² – (2y)²]

= 3(3x – 2y)(3x + 2y)

Question: 3

Solve:

144a² – 289b²

Solution:

144a² – 289b²

= (12a)² – (17b)²

= (12a – 17b)(12a + 17b)

Question: 4

Solve:

12m² – 27

Solution:

12m² – 27

= 3(4m² – 9)

= 3[(2m)² – 3²]

= 3(2m – 3)(2m + 3)

Question: 5

Solve:

125x² – 45y²

Solution:

125x² – 45y²

= 5(25x² – 9y²)

= 5[(5x)² – (3y)²]

= 5(5x – 3y)(5x + 3y)

Question: 6

Solve:

144a² – 169b²

Solution:

144a² – 169b²

= (12a)² – (13b²)

= (12a – 13b)(12a + 13b)

Question: 7

Solve:

(2a – b)² – 16c²

Solution:

(2a – b)² – 16c²

= (2a – b)² – (4c)²

= [(2a – b) –  4c][(2a – b) + 4c]

= (2a – b – 4c)(2a – b + 4c)

Question: 8

Solve:

(x + 2y)² – 4(2x – y)²

Solution:

(x + 2y)² – 4(2x – y)² = (x + 2y)² – [2(2x – y)]²

= [(x + 2y) – 2(2x – y)][(x + 2y) + 2(2x – y)]

= (x + 2y – 4x + 2y)(x + 2y + 4x – 2y)

= 5x(4y – 3x)

Question: 9

Solve:

3a⁵ – 48a³

Solution:

3a⁵ – 48a³

= 3a³(a² – 16)

= 3a³(a² – 4²)

= 3a³(a – 4)(a + 4)

Question: 10

Solve:

a⁴ – 16b⁴

Solution:

a⁴ – 16b⁴

= a⁴ – 2⁴b⁴

= (a²)² – (2²b²)²

= (a² – 2²b²)(a² + 2²b²)

= [a² – (2b)²](a² + 4b²)

(a – 2b)(a + 2b)(a² + 4b²)

Question: 11

Solve:

 x⁸– 1

Solution:

x⁸– 1 = (x⁴)² – 1²

= (x⁴ – 1)(x⁴ + 1)

= [(x²)² – 1²](x⁴ + 1)

= (x² – 1)(x² + 1)(x⁴ + 1)

= (x² – 1²)(x² + 1)(x⁴ + 1)

= (x – 1)(x +1)(x² + 1)(x⁴ + 1)

Question: 12

Solve:

64 – (a + 1)²

Solution:

64 – (a + 1)²

= (8)² – (a + 1)²

= [8 – (a + 1)][8 + (a + 1)]

= (8 – a – 1)(8 + a + 1)

= (7 – a) (9 + a)

Question: 13

Solve:

36L² – (m + n)²

Solution:

36L² – (m + n)²

= (6L)² – (m + n)²

= [6L – (m +n)][6L + (m +n)]

= (6L – m – n)(6L + m + n)

Question: 14

Solve:

25x⁴y⁴ – 1

Solution:

25x⁴y⁴ – 1

= (5x²y²)² – 1

= (5x²y² – 1)(5x²y² + 1)

Question: 15

Solve:

a⁴ – 1/b⁴

Solution:

a⁴ – 1/b⁴

= (a²)² – 1/(b²)²

= a² – 1/b²a² + 1/b²

= a – 1/ba + 1/ba² + 1/b²

Question: 16

Solve:

x³  – 144x

Solution:

x³  – 144x

= x(x² – 144)

= x(x² – 12²)

= x(x – 12)(x + 12)

Question: 17

Solve:

 (x – 4y)² – 625

Solution:

(x – 4y)² – 625

= (x – 4y)² – 25²

= [(x – 4y) – 25][(x – 4y) + 25]

= (x – 4y – 25)(x – 4y + 25)

Question: 18

Solve:

9(a – b)² – 100(x – y)²

Solution:

9(a – b)² – 100(x – y)²

= [3(a – b)]² – [10(x – y)]²

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

= (3a – 3b – 10x + 10y)(3a – 3b + 10x – 10y)

Question: 19

Solve:

(3 + 2a)² – 25a²

Solution:

(3 + 2a)² – 25a²

= (3 + 2a)² – (5a)²

= [(3 + 2a) – 5a][(3 + 2a) + 5a]

= (3 + 2a – 5a)(3 + 2a + 5a)

= (3 – 3a)(3 + 7a)

= 3(1 - a)(3 + 7a)

Question: 20

Solve:

(x + y)² – (a – b)²

Solution:

(x + y)² – (a – b)²

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

= (x + y – a + b)(x + y + a – b)

Question: 21

Solve:

Solution:

Question: 22

Solve:

75a³b² – 108ab⁴

Solution:

75a³b² – 108ab⁴

= 3ab²(25a² – 36b²)

= 3ab² [(5a)² – (6b)²]

= 3ab² (5a – 6b) (5a + 6b)

Question: 23

Solve:

 x⁵ – 16x³

Solution:

x⁵ - 16x³

= x³(x² – 16)

= x³(x² – 4²)

= x³(x – 4)(x + 4)

Question: 24

Solve:

Solution:

Question: 25

Solve:

256x⁵ – 81x

Solution:

256x⁵ – 81x

= x(256x⁴ – 81)

= x[(16x²)² – 9²]

= x(16x² + 9)(16x² – 9)

= x(16x² + 9)[(4x)² – 3²]

= x(16x² + 9)(4x + 3)(4x – 3)

Question: 26

Solve:

a⁴- (2b + c)⁴

Solution:

a⁴ – (2b + c)⁴

= (a²)² – [(2b + c)²]²

= [a² + (2b + c)²][a² – (2b + c)²]

= [a² + (2b + c)²]{[a + (2b + c)][a – (2b + c)]}

= [a² + (2b + c)²](a + 2b + c)(a – 2b – c)

Question: 27

Solve:

(3x + 4y)⁴ – x⁴

Solution:

(3x + 4y)⁴ – x⁴

= [(3x + 4y)²]² – (x²)²

= [(3x + 4y)² + x²][(3x + 4y)² – x²]

= [(3x + 4y)² + x²][(3x + 4y) + x] [(3x + 4y) – x]

= {(3x + 4y)² + x²} (3x + 4y + x)(3x + 4y – x)

= {(3x + 4y)² + x²} (4x + 4y)(2x + 4y)

= {(3x + 4y)² + x²} 4(x + y) 2 (x + 2y)

= 8{(3x + 4y)² + x²} (x + y)(x + 2y)

Question: 28

Solve:

p²q² – p⁴q⁴

Solution:

p²q² – p⁴q⁴

= p²q²(1 – p²q²)

= p²q² [1 – (pq)²]

= p²q² (1 – pq)(1 + pq)

Question: 29

Solve:

3x³y – 243xy³

Solution:

3x³y – 243xy³

= 3xy(x² – 81y²)

= 3xy[x² – (9y)²]

= 3xy(x – 9y)(x + 9y)

Question: 30

Solve:

a⁴b⁴ – 16c⁴

Solution:

a⁴b⁴ – 16c⁴

= [(a²b²)² – (4c²)²]

= (a²b² + 4c²)(a²b² – 4c²)

= (a²b² + 4c²)[(ab)² – (2c)²]

= (a²b² + 4c²)(ab + 2c)(ab – 2c)

Question: 31

Solve:

x⁴ – 625

Solution:

x⁴ – 625

= (x²)² – 25²

= (x² + 25)(x² – 25)

= (x²+ 25)(x² – 5²)

= (x² + 25)(x + 5)(x – 5)

Question: 32

Solve:

x⁴ – 1

Solution:

x⁴ – 1

= (x²)² – 1

= (x² + 1)(x² – 1)

= (x² + 1)(x + 1)(x – 1)

Question: 33

Solve:

49(a – b)² – 25(a + b)²

Solution:

49(a – b)² – 25(a + b)²

= [7(a – b)²] – [5(a + b)]²

= [7(a – b) – 5(a + b)] [7(a – b) + 5(a + b)]

= (7a – 7b – 5a – 5b)(7a – 7b + 5a + 5b)

= (2a – 12b)(12a – 2b)

= 2(a – 6b) 2(6a – b)

= 4(a – 6b)(6a – b)

Question: 34

Solve:

x – y – x² + y²

Solution:

x – y – x² + y²

= (x – y) + (y² – x²)

= (x – y) + (y + x)(y – x)

= (x – y) – (y + x)(x – y)   [since, (y – x) = – (x – y)]

= (x – y)[1 – (y + x)]

= (x – y)(1 – x – y)

Question: 35

Solve:

16(2x – 1)² – 25y²

Solution:

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)

= (8x – 5y – 4)(8x + 5y – 4)

Question: 36

Solve:

 4(xy + 1)² – 9(x – 1)²

Solution:

4(xy + 1)² – 9(x – 1)²

= [2(xy + 1)]² – [3(x – 1)]²

= [2(xy + 1) – 3(x – 1)] [2(xy + 1) + 3(x – 1)]

= (2xy + 2 – 3x + 3)(2xy + 2 + 3x – 3)

= (2xy – 3x + 5)(2xy + 3x – 1)

Question: 37

Solve:

(2x + 1)² – 9x⁴

Solution:

(2x + 1)² – 9x⁴

= (2x + 1)² – (3x²)²

= [(2x + 1) – 3x²][(2x + 1) + 3x²]

= (-3x² + 2x + 1)(3x² + 2x + 1)

= (-3x² + 3x – x + 1)(3x² + 2x + 1)

= {3x(-x + 1) + 1(-x + 1)}(3x² + 2x + 1)

= (-x + 1)(3x + 1)(3x² + 2x + 1)

= – (x – 1)(3x + 1)(3x² + 2x + 1)

Question: 38

Solve:

 x⁴ – (2y – 3z)²

Solution:

x⁴ – (2y – 3z)²

= (x²)² – (2y – 3z)²

= [x² – (2y – 3z)][x² + (2y – 3z)]

= (x² – 2y + 3z)(x² + 2y – 3z)

Question: 39

Solve:

a² – b² + a – b

Solution:

a² – b² + a –b = (a² – b²) + (a – b)

= (a + b)(a – b) + (a – b)

= (a – b)(a + b + 1)

Question: 40

Solve:

16a⁴ – b⁴

Solution:

16a⁴ – b⁴

= (4a²)² – (b²)²

= (4a² + b²)(4a² – b²)

= (4a² + b²)[(2a)² – b²]

= (4a² + b²)(2a + b)(2a – b)

Question: 41

Solve:

a⁴ – 16(b – c)⁴

Solution:

a⁴ – 16(b – c)⁴

= (a²)² – [4(b – c)²]²

= [a² + 4(b – c)²][a² – 4(b – c)²]

= [a² + 4(b – c)²] [a² – [2(b – c)]²]

= [a² + 4(b – c)²][a + 2(b – c)][a – 2(b – c)]

= [a² + 4(b – c)²](a + 2b – 2c)(a – 2b + 2c)

Question: 42

Solve:

2a⁵ – 32a

Solution:

2a⁵ – 32a

= 2a(a⁴ – 16)

= 2a[(a²)² – 4²]

= 2a(a² + 4)(a² – 4)

= 2a(a² + 4)(a² – 2²)

= 2a(a² + 4)(a + 2)(a – 2)

= 2a(a – 2)(a + 2)(a² + 4)

Question: 43

Solve:

a⁴b⁴ – 81c⁴

Solution:

a⁴b⁴ – 81c⁴

= (a²b²)² – (9c²)²

= (a²b² + 9c²)(a²b² – 9c²)

= (a²b² + 9c²)[(ab)² – (3c)²]

= (a²b²+ 9c²)(ab + 3c)(ab – 3c)

Question: 44

Solve:

xy⁹ – yx⁹

Solution:

xy⁹ – yx⁹

= xy(y⁸ – x⁸)

= xy[(y⁴)² – (x⁴)²]

= xy(y⁴ + x⁴)[(y²)² – (x²)²]

= xy(y⁴ + x⁴)(y² + x²)(y² – x²)

=  xy(y⁴ + x⁴)(y² + x²)(y + x)(y – x)

Question: 45

Solve:

x³ – x

Solution:

x³ – x = x(x² – 1)

= x(x – 1)(x + 1)

Question: 46

Solve:

18a²x² – 32

Solution:

18a²x² – 32

= 2(9a²x² – 16)

= 2[(3ax)² – 4²]

= 2(3ax – 4)(3ax + 4)