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Question
Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD
(x3 – 1) (x + 1), (x3 + 1)
Solution
p(x) = (x3 – 1) (x + 1) = (x – 1) (x2 + x + 1) (x + 1)
g(x) = x3 + 1 = (x + 1) (x2 – x + 1)
G.C.D = (x + 1)
L.C.M = (x + 1) (x – 1) (x2 + x + 1) (x2 – x + 1)
L.C.M × G.C.D = (x + 1) (x – 1) (x2 + x + 1) (x2 – x + 1) × (x + 1)
= (x + 1)2 (x – 1) (x2 + x + 1) (x2 – x + 1) ...(1)
p(x) × g(x) = (x – 1) (x2 + x + 1) (x + 1) (x + 1) (x2 – x + 1)
= (x + 1)2 (x – 1) (x2 + x + 1) (x2 – x + 1) ...(2)
From (1) and (2) we get
L.C.M. × G.C.D. = p(x) × g(x)
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