Unit 4: Factorization - Solved Exercise 4.5 - Mathematics For Class IX (Science Group)

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Unit 4: Factorization
Solved Exercise 4.5

1. Factorize the following:
(i) x³ + 8y³
Solution:
⇒ x³ + 8y³
⇒ (x)³ + (2y)³
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²) ]
⇒ (x + 2y)[(x)² - (x)(2y) + (2y)²]
⇒ (x + 2y)(x² - 2xy + 4y²)
Therefore,
⇒ x³ + 8y³ = (x + 2y)(x² - 2xy + 4y²) Ans.

(ii) a¹¹ + a²b⁹
Solution:
⇒ a¹¹ + a²b⁹
By Taking a² as a common from the expression,
⇒ a²(a⁹ + b⁹)
⇒ a²{(a³)³ + (b³)³)
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²) ]
⇒ a²{(a³ + b³)[(a³)² - (a³)(b³) + (b³)²]}
⇒ a²(a³ + b³)(a⁶ - a³b³ + b⁶)
⇒ a²{(a)³ + (b)³)}(a⁶ - a³b³ + b⁶)
Again By Using Formula:
[ a³ + b³ = (a + b)(a² - ab + b²) ]
⇒ a²{(a + b)[(a)² - (a)(b) + (b)²]}(a⁶ - a³b³ + b⁶)]
⇒ a²{(a + b)(a² - ab + b²)(a⁶ - a³b³ + b⁶)
⇒ a²(a + b)(a² - ab + b²)(a⁶ - a³b³ + b⁶)
Therefore,
⇒ a¹¹ + a²b⁹ = a²(a + b)(a² - ab + b²)(a⁶ - a³b³ + b⁶) Ans.

(iii) a⁶ + 1
Solution:
⇒ a⁶ + 1
⇒ {(a²)³ + (1)³}
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²) ]
⇒ (a² +1)[(a²)² - (a²)(1) + (1)²]
⇒ (a² + 1)(a⁴ - a² + 1
Therefore,
⇒ a⁶ + 1 = (a² + 1)(a⁴ - a² + 1

(iv) a³b³ + 512
Solution:
⇒ (ab)³ + (8)³
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²)]
⇒ (ab +8)[(ab)² - (ab)(8) + (8)²]
⇒ (ab + 8)(a²b² - 8ab + 64
Therefore,
⇒ a³b³ + 512 = (ab + 8)(a²b² - 8ab + 64)

(v) a³b³ + 27b⁶
Solution:
⇒ (ab)³ + (3b)³
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²)]
⇒ (ab +3b)[(ab)² - (ab)(3b) + (3b)²]
⇒ (ab + 3b)(a²b² - 3ab² + 9b²
Therefore,
⇒ a³b³ + 27b⁶ = (ab + 3b)(a²b² - 3ab² + 9b²)

(vii) x⁹ + x³y⁶z⁹
Solution:
⇒ x⁹ + x³y⁶z⁹
By Taking x³ as a common from the expression,
⇒ x³(x⁶ + y⁶z⁹)
⇒ x³{(x²)³ + (y²z³)³}
By Using Formula:
[a³ + b³ = (a + b)(a² - ab + b²)]
⇒ x³(x² + y²z³)[(x²)² - (x²)(y²z³) + (y²z³)²]
⇒ x³(x² + y²z³)(x⁴ - x²y²z³ + y⁴z⁶)
Therefore,
⇒ x⁹ + x³y⁶z⁹ = x³(x² + y²z³)(x⁴ - x²y²z³ + y⁴z⁶) Ans.