Mathematics
Factorise each of the following:
(i) 27y3 + 125z3
(ii) 64m3 - 343n3
Polynomials
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Answer
(i) 27y3 + 125z3
[∵ x3 + y3 = (x + y)(x2 - xy + y2)]
⇒ 27y3 + 125z3 = (3y)3 + (5z)3
Putting x = 3y, y = 5z
R.H.S = (x + y)(x2 - xy + y2)
= (3y + 5z)[(3y)2 - (3y)(5z) + (5z)2]
= (3y + 5z)(9y2 - 15yz + 25z2)
Hence, 27y3 + 125z3 = (3y + 5z)(9y2 - 15yz + 25z2)
(ii) 64m3 - 343n3
[∵ a3 - b3 = (a - b)(a2 + ab + b2)]
⇒ 64m3 - 343n3 = (4m)3 - (7n)3
Putting a = 4m, b = 7n
R.H.S = (a - b)(a2 + ab + b2)
= (4m - 7n)[(4m)2 + (4m)(7n) + (7n)2]
= (4m - 7n)(16m2 + 28mn + 49n2)
Hence, 64m3 - 343n3 = (4m - 7n)(16m2 + 28mn + 49n2)
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Related Questions
Factorise each of the following:
(i) 8a3 + b3 + 12a2b + 6ab2
(ii) 8a3 - b3 - 12a2b + 6ab2
(iii) 27 - 125a3 - 135a + 225a2
(iv) 64a3 - 27b3 - 144a2b + 108ab2
(v) 27p3 - - p2 + p
Verify :
(i) x3 + y3 = (x + y) (x2 - xy + y2)
(ii) x3 - y3 = (x - y) (x2 + xy + y2)
Factorise : 27x3 + y3 + z3 - 9xyz
Verify that x3 + y3 +z3 - 3xyz = (x + y + z)[(x - y)2 + (y - z)2 + (z - x)2]