Question

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD

(x³ – 1) (x + 1), (x³ + 1)

Answer 1

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)