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Problems on Factorization of Expressions of the Form a\(^{2}\) - b\(^{2}\)
Here we will solve different types of Problems on Factorization of expressions of the form a2 – b2.
1. Resolve into factors: 49a2 – 81b2
Solution:
Given expression = 49a2 – 81b2
= (7a)2 – (9b)2
= (7a + 9b)(7a – 9b).
2. Factorize: (x + y)2 – 4(x – y)2
Solution:
Given expression = (x + y)2 – 4(x – y)2
= (x + y)2 – {2(x – y)}2
= {(x + y) + 2(x – y)}{(x + y) - 2(x – y)}
= (x + y + 2x – 2y)(x + y - 2x + 2y)
= (3x – y)(3y - x).
3. Factorize the expression (x2 + y2 – z2)2 – 4x2y2 of the form a2 – b2.
Solution:
Given expression = (x\(^{2}\) + y\(^{2}\) – z\(^{2}\))\(^{2}\) – 4x\(^{2}\)y\(^{2}\)
= (x\(^{2}\) + y\(^{2}\) – z\(^{2}\))\(^{2}\) – (2xy)\(^{2}\)
= (x\(^{2}\) + y\(^{2}\) – z\(^{2}\) + 2xy)(x\(^{2}\) + y\(^{2}\) – z\(^{2}\) – 2xy)
= (x\(^{2}\) + 2xy + y\(^{2}\) - z\(^{2}\))(x\(^{2}\) – 2xy + y\(^{2}\) – z\(^{2}\))
= {(x\(^{2}\) + 2xy + y\(^{2}\)) - z\(^{2}\)}{(x\(^{2}\) - 2xy + y\(^{2}\)) – z\(^{2}\)}
= {(x + y)\(^{2}\) - z\(^{2}\)}{(x - y)\(^{2}\) - z\(^{2}\)}
= (x + y + z)(x + y - z)(x - y + z) (x - y - z).
4. Factorize 2x\(^{2}\) - 18 of the form a\(^{2}\) – b\(^{2}\).
Solution:
Given expression = 2x\(^{2}\) - 18
= 2(x\(^{2}\) – 9)
= 2(x\(^{2}\) – 3\(^{2}\))
= 2(x + 3)(x - 3)
5. Factorize: 25(a + b)\(^{2}\) – (a – b)\(^{2}\)
Solution:
Given expression = 25(a + b)\(^{2}\) – (a – b)\(^{2}\)
= {5(a + b)}\(^{2}\) – (a – b)\(^{2}\)
= {5(a + b) + (a – b)}{5(a + b) – (a – b)}
= (5a + 5b + a – b)(5a + 5b - a + b)
= (6a + 4b)(4a + 6b)
= {2(3a + 2b)}{2(2a + 3b)}
= 4(3a + 2b)(2a + 3b)
6. Factorize the expression 9(x + y)\(^{2}\) – x\(^{2}\) of the form a\(^{2}\) – b\(^{2}\).
Solution:
Given expression = 9(x + y)\(^{2}\) – x\(^{2}\)
= {3(x + y)}\(^{2}\) – x\(^{2}\)
= {3(x + y) + x}{3(x + y) - x}
= (3x + 3y + x)(3x + 3y - x)
= (4x + 3y)(2x + 3y)
7. Factorize the expression 9x\(^{2}\) – 4(y + 2x)\(^{2}\) of the form a\(^{2}\) – b\(^{2}\).
Solution:
Given expression = 9x\(^{2}\) – 4(y + 2x)\(^{2}\)
= (3x)\(^{2}\) – {2(y + 2x)}\(^{2}\)
= {3x + 2(y + 2x)}{3x - 2(y + 2x)}
= (3x + 2y + 4x)(3x - 2y - 4x)
= (7x + 2y)(-x - 2y)
= (7x + 2y){-(x + 2y)}
= -(7x + 2y)(x + 2y)
8. Factorize: 1 – (b – c)\(^{2}\)
Solution:
Given expression = 1 – (b – c)\(^{2}\)
= 1\(^{2}\) – (b – c)\(^{2}\)
= {1 + (b – c)}{1 - (b – c)}
= (1 + b – c)(1 - b + c)
9. Factorize: 81a\(^{4}\) – 16b\(^{4}\)
Solution:
Given expression = 81a\(^{4}\) – 16b\(^{4}\)
= (9a\(^{2}\))\(^{2}\) – (4b\(^{2}\))\(^{2}\)
= (9a\(^{2}\) + 4b\(^{2}\))(9a\(^{2}\) - 4b\(^{2}\))
= (9a\(^{2}\) + 4b\(^{2}\)){(3a)\(^{2}\) - (2b)\(^{2}\)}
= (9a\(^{2}\) + 4b\(^{2}\))(3a + 2b)(3a - 2b)
10. Factorize: 16ax\(^{4}\) – ay\(^{4}\)
Solution:
Given expression = 16ax\(^{4}\) – ay\(^{4}\)
= a(16x\(^{4}\) – y\(^{4}\))
= a{(4x\(^{2}\))\(^{2}\) – (y\(^{2}\))\(^{2}\)}
= a(4x\(^{2}\) + y\(^{2}\))(4x\(^{2}\) - y\(^{2}\))
= a(4x\(^{2}\) + y\(^{2}\)){(2x)\(^{2}\) - y\(^{2}\)}
= a(4x\(^{2}\) + y\(^{2}\))(2x + y)(2x – y)
11. Factorize: a\(^{4}\) – 16b\(^{4}\)
Solution:
Given expression = a\(^{4}\) – 16b\(^{4}\)
= (a\(^{2}\))\(^{2}\) – (4b\(^{2}\))\(^{2}\)
= (a\(^{2}\) + 4b\(^{2}\))(a\(^{2}\) - 4b\(^{2}\))
= (a^2 + 4b\(^{2}\)){a\(^{2}\) – (2b)\(^{2}\)}
= (a^2 + 4b\(^{2}\))(a + 2b)(a - 2b)
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