RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5

These Solutions are part of RD Sharma Class 8 Solutions. Here we have given RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5

Other Exercises

Factorize each of the following expressions :
Question 1.
16x²-25y²Solution:
16x² – 25y² = (4x)² – (5y)²{∵ a² – b² = (a + b) (a – b)}
= (4x + 5y) (4x – 5y)

Question 2.
27x² – 12y²Solution:
27x² – 12y² = 3 (9x² – 4y²) {∵ a² -b² = (a + b) (a – b)}
= 3 [(3x)² – (2y)²]
= 3 (3x + 2y) (3x – 2y)

Question 3.
144a²– 289b²
Solution:
144a² – 289b² = (12a)² – (17b)²{ ∵ a² – b² = (a + b) (a – b}
= (12a+ 17b) (12a- 17b)

Question 4.
12m² – 27
Solution:
12m² – 27 = 3 (4m² – 9)
= 3 {(2m)²-(3)²} {∵ a² – b² = (a + b) (a – b)}
= 3 (2m + 3) (2m – 3)

Question 5.
125x² – 45y²
Solution:
125x² – 45y² = 5 (25x² – 9y²)
= 5 {(5x-)² – (3y)²} {∵ a² – b² = (a + b) (a – b}
= 5 (5x + 3y) (5x – 3y)

Question 6.
144a² – 169b²
Solution:
144a² – 169b² = (12a)² – (13b)²{∵ a² -b² = (a + b) (a – b)}
= (12a + 13b) (12a-13b)

Question 7.
(2a – b)² – 16c²
Solution:
(2a – b)² – 16c² = (2a – b)² – (4c)²{∵ a² – b² = (a + b) (a – b)}
= (2a – b + 4c) (2a – b – 4c)

Question 8.
(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 – (4x – 2y)²{∵ a²– b² = (a + b) (a – b)}
= (a + 2y + 4x – 2y) (x + 2y – 4x + 2y)
= 5x (-3x + 4y)

Question 9.
3a⁵ – 48a³
Solution:
3a⁵ – 48a³ = 3a³ (a²– 16)
= 3a³ {(a)² – (4)²} {∵ a² – b² = (a + b) (a – b)}
= 3a³ (a + 4) (a – 4)

Question 10.
a⁴ – 16b⁴Solution:
a⁴ – 16b⁴ = (a²)² – (4b²)²= (a² + 4b²) (a² – 4b²)
= (a² + 4b²) {(a)² – (2b)² } { ∵ a² – b² = (a + b) (a – b)}
= (a² + 4b²) (a + 2b) (a – 2b)

Question 11.
x⁸ – 1
Solution:
x⁸ – 1 = (x⁴)² – (1)²= (x⁴ + 1) (x⁴ – 1)
= (x⁴+ 1) I (x²)² – (1)²} {∵ a² – b² = (a + b) (a – b)}
= (x⁴ + 1) (x² + 1) (x² – 1)
= (x⁴ + 1) (x² + 1) {(x)² – (1)²}
= (x⁴+ 1)(x² + 1)(x+ 1)(x- 1)
= (x-1)(x+ 1) (x² + 1) (x⁴ + 1)

Question 12.
64 – (a + 1)²Solution:
64 – (a + 1)² = (8)² – (a + 1)²{∵ a² – b² = (a + b) (a – b)}
= (8 + a + 1) (8 – a – 1)
= (9 + a) (7 – a)

Question 13.
36l² – (m + n)²Solution:
36l² – (m + n)² = (6l)² – (m + n)²{∵ a² – b² = (a + b) (a – b)}
= (6l + m + n) (6l – m – n)

Question 14.
25x⁴y⁴ – 1
Solution:
25x⁴y⁴ – 1 = (5x⁴y⁴)² – (1)²{ ∵ a² – b² = (a + b) (a – b)}
= (5x⁴y⁴ + 1) (5x²y² – 1)

Question 15.
RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 1.1
Solution:

Question 16.
x³ – 144x
Solution:
x³ – 144x = x (x² – 144)
= x {(x)² – (12)²} {∵ a² – b² = (a + b) (a – b)}
= x (x + 12) (x – 12)

Question 17.
(x – 4y)² – 625
Solution:
(x – 4y)² – 625
= (x – 4y)² – (25)²{∵ a² – b² = (a + b) (a – b)}
= (x – 4y + 25) (x -4y – 25)

Question 18.
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{∵ a² – b² = (a + b) (a – b)}
= (3a – 3b)² – (10x – 10y)²= (3a – 3b + 10x – 10y) (3a – 3b – 10x + 10y)

Question 19.
(3 + 2a)² – 25a²
Solution:
(3 + 2a)² – 25a²= (3 + 2a)² – (5a)²(∵ a² – b² = (a + b) (a – b)}
= (3 + 2a + 5a) (3 + 2a – 5a)
= (3 + 7a) (3 – 3a)
= (3 + 7a) 3 (1 – a)
= 3(1-a) (3 +7a)

Question 20.
(x + y)² – (a – b)²
Solution:
RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 3

Question 21.
RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 4
Solution:

Question 22.
75a³b² – 108ab⁴
Solution:
75a³b² – 108ab⁴= 3ab² (25a² – 36b²)
= 3ab² {(5a)² – (6b)²} {∵ a² – b² = (a + b) (a – b)}
= 3ab² (5a + 6b) (5a – 6b)

Question 23.
x⁵– 16x³
Solution:
x⁵ – 16x³ = x³ (x² – 16)
= x³ {(x)² – (4)²} {∵ a² – b² = (a + b) (a – b)}
= x³ (x + 4) (x – 4)

Question 24.
RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 6
Solution:

Question 25.
256x⁵ – 81x
Solution:
256x⁵– 81x = x(256x⁴– 81)
= x {(16x²)² – (9)²} {∵ a² – b² = {a + b) (a – b)}
= x (16x² + 9) (16x² – 9)
= x (16x² + 9) {(4x)² – (3)²}
= x (16x² + 9) (4x + 3) (4x-3)

Question 26.
a⁴ – (2b + c)⁴
Solution:
a⁴ – (2b + c)⁴
= (a²)² – [(2b + c)²]²{∵ a² – b² = (a + b) (a – b)}
= {a² + (2b + c)²} {a² – (2b + c)²}
= {a² + (2b + c)²} {(a)² – (2b + c)²}
= {a² + (2b + c)²} (a + 2b + c) (a -2b- c)

Question 27.
(3x + 4y)⁴ – x⁴
Solution:
(3x + 4y)⁴ – x⁴ – [(3x + 4y)²]² – (x²)²= [(3x + 4y)² + x²] [(3x + 4y)² – x²] {∵ a² – b² = (a + b) (a – b)
= [(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 (x + y) (x + 2y) [(3x + 4y)² + x²]

Question 28.
p²q² – p⁴q⁴
Solution:
p²q²– p⁴q⁴=p²q² (1 -p²q²)
=p²q² [(1)² – (pq)²] {∵ a² – b² = (a + b) (a – b)
= p²q² (1 +pq) (1 -pq)

Question 29.
3x³y – 243xy³Solution:
3x³y – 243xy³
= 3xy (x² – 81y²)
= 3xy [(x)² – (9y)²]
= 3xy (x + 9y) (x – 9y)

Question 30.
a⁴b⁴ – 16c⁴
Solution:
a⁴b⁴ – 16c⁴ = (a²b²)² – (4c²)²= (a²b² + 4c²) (a²b² – 4c²)
= (a²b² + 4c²) [(ab)² – (2c)²] {∵ a² – b² = (a + b) (a – b)
= (a²b² + 4c²) (ab + 2c) (ab – 2c)

Question 31.
x⁴-625
Solution:
x⁴ – 625 = (x²)² – (25)²{∵ a² – b² – (a + b) (a – b)
= (x² + 25) (x² – 25)
= (x² + 25) [(x)² – (5)²]
= (x² + 25) (x + 5) (x – 5)

Question 32.
x⁴-1
Solution:
x⁴ – 1 = (x²)² – (1)² = (x² + 1) (x² – 1)
= (x² + 1) [(x)² – (1)²]
= (x² + 1) (x + 1) (x – 1)

Question 33.
49 (a – b)² -25 (a + b)²
Solution:
49 (a – by -25 (a + b)²= [7 (a – b)]² – [5 (a + b)]²= (7a – 7b)² – (5a + 5b)²{∵ a² – b² = (a + b) (a – b)
= (7a -7b + 5a + 5b) (7a – 7b -5a- 5b)
=(12a – 2b)(2a – 12b)
= 2 (6a – b) 2 (a – 6b)
= 4 (6 a- b) (a – 6b)

Question 34.
x – y – x² + y²
Solution:
x-y-x² + y²= (x-y)-(x²-y²) {∵ a² – b² = (a + b) (a – b)
= {x-y)-(x + y)(x-y)
= (x-y)(1 – x – y)

Question 35.
16 (2x – 1)² – 25y²Solution:
16 (2x – 1)² – 25y²
= [4 (2x – 1)]² – (5y)²= (8x – 4)² – (5y)²
= (8x – 4 + 5y) (8x -4-5y)
= (8x + 5y – 4) (8x – 5y – 4)

Question 36.
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= (2xy + 2)² – (3x – 3)²{∵ a² – b² = (a + b) (a – b)
= (2xy + 2 + 3x – 3) (2xy + 2 – 3x + 3)
= (2xy + 3x – 1) (2xy – 3x + 5)

Question 37.
(2x + 1)² – 9x⁴Solution:
(2x + 1)² – 9x⁴ = (2x + 1)² – (3x²)²{∵ a² – b² = (a + b) (a – b)
= (2x + 1 + 3x²) (2x + 1 – 3x²)
= (3x² + 2x + 1) (-3x + 2x + 1)

Question 38.
x⁴ – (2y- 3z)²
Solution:
x⁴ – (2y – 3z)² = (x²)² – (2y – 3z)²
= (x² + 2y- 3z) (x² – 2y + 3z)

Question 39.
a²-b² +a-b
Solution:
a² – b² + a – b
= (a + b) {a – b) + 1 (a – b)
= (a – b) (a + b + 1)

Question 40.
16a⁴ – b⁴
Solution:
16a⁴ – b⁴
= (4a²)² – (b²)²{ ∵ a² – b² = (a + b) (a – b)
= (4a² + b²) (4a² – b²)
= (4a² + b²) {(2a)² – (b)²}
= (4a² + b²) (2a + b) (2a – b)

Question 41.
a⁴ – 16 (b – c)⁴Solution:
a⁴ – 16 (b- c)⁴ = (a²)² – [4 (b – c)²]²{ ∵ a² – b² = (a + b) (a – b)
= [a² + 4 (b – c)²] [a² – 4 (b – c)²]
= [a² + 4 (b – c)²] [(a)² – [2 (b – c)]²]
= [a² + 4 (b – c)²] [(a)² – (2b – 2c)²]
= [a² + 4 (b – c)²] (a + 2b – 2c) (a – 2b + 2c)

Question 42.
2a⁵ – 32a
Solution:
2a⁵ – 32a = 2a (a⁴ – 16)
= 2a [(a²)² – (4)²] {∵ a² – b² = (a + b) (a – b)
= 2a (a² + 4) (a² – 4)]
= 2a (a² + 4) [(a)² – (2)²]
= 2a (a² + 4) (a + 2) (a – 2)

Question 43.
a⁴b⁴ – 81c⁴
Solution:
a⁴b⁴ – 81c⁴ = (a²b²)² – (9c²)²= (a²b² + 9c²) (a²b² – 9c²) {∵ a² – b² = (a + b) (a – b)
= (a²b² + 9c²) {(ab)² – (3c)²}
= (a²b² + 9c²) (ab + 3c) (ab – 3c)

Question 44.
xy⁹-yx⁹
Solution:
xy⁹ – yx⁹ = xy (y⁸ – x⁸)
= xy [(y⁴)² – (x⁴)²] {∵ a² – b² = (a + b) (a – b)}
= xy(y⁴ + x⁴)(y⁴-x⁴)
= 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.
x³ -x
Solution:
x³-x = x(x²– 1)
= x [(x)² – (1)²] = x (x + 1) (x – 1)

Question 46.
18a²x² – 32
Solution:
18a²x² – 32
= 2 [9a²x² – 16]
= 2 [(3ax)² – (4)²] {∵ a² – b² = (a + b) (a – b)
= 2 (3ax + 4) (3ax – 4)

Hope given RD Sharma Class 8 Solutions Chapter 7 Factorizations Ex 7.5 are helpful to complete your math homework.

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