Unit 4: Factorization - Solved Exercise 4.4 - Mathematics For Class IX (Science Group)

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Unit 4: Factorization
Solved Exercise 4.4

1. Factorize the following:
(i) b³ + 3b²c + 3bc² + c³
Solution:
⇒ b³ + 3b²c + 3bc² + c³
⇒ (b)³ + 3(b)²(c) + 3(b)(c)² + (c)³
By using formula:
[∵ (a³ + 3a²b + 3ab² + b³) = (a + b)³]
⇒ (b + c)³ Ans.

(ii) 8x³ + 12x²y + 6xy² + y³
Solution:
⇒ 8x³ + 12x²y + 6xy² + y³

 (Rough Work: 8 = 2 x 2 x 2 = 2³)

⇒ 2³x³ + 12x²y + 6xy² + c³
⇒ (2x)³ + 3(2x)²)(y) + 3(2x)(y)² + (y)³
By using formula:
[∵ (a³ + 3a²b + 3ab² + b³) = (a + b)³]
⇒ (2x + y)³ Ans.

(iv) 8x³ + 36x² + 54x + 27
Solution:
⇒ 8x³ + 36x² + 54x + 27

 (Rough Work: 8 = 2 x 2 x 2 = 2³ and  27 = 3 x 3 x 3 = 3³)

⇒ 2³x³ + 36x² + 54x + 3³
⇒ (2x)³ + 3(2x)²)(3) + 3(2x)(3)² + (3)³
By using formula:
[∵ (a³ + 3a²b + 3ab² + b³) = (a + b)³]
⇒ (2x + 3)³ Ans.

2. Find The Factors Of:
(i) d³ - 6d²c + 12dc² - 8c³
Solution:
⇒ d³ - 6d²c + 12dc² - 8c³

 (Rough Work: 8 = 2 x 2 x 2 = 2³)

⇒ (d)³ + 3(d)²(-2c) + 3(d)(-2c)² + (-2c)³
⇒ (d)³ - 3(d)²(2c) + 3(d)(2c)² - (2c)³
By using formula:
[∵ (a³ - 3a²b + 3ab² - b³) = (a - b)³]
⇒ (d - 2c)³ Ans.

(vi) 125z³ - 75z²y² + 15zy⁴ - y⁶
Solution:
⇒ 125z³ - 75z²y² + 15zy⁴ - y⁶

 Rough Work:
125 = 5 x 5 x 5 = 5³

⇒ 5³z³ - 75z²y² + 15zy⁴ + (-y²)³
⇒ (5z)³ + 3(5z)²)(-y²) + 3(5z)(-y²)² + (-y²)³
By using formula:
[∵ (a³ - 3a²b + 3ab² - b³) = (a - b)³]
⇒ (5z - y²)³ Ans.