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Wednesday, 18 September 2024

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.





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