InterviewSolution
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InterviewSolution
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Find the LCM and GCD for the following and verify that f(x) x g(x) = LCM x GCD. (i) 21x2y, 35xy2 (ii) (x3 – 1)(x + 1), x3 + 1 (iii) (x3 – 1) (x + 1), (x3 – 1) (iv) (x2 y + xy2), (x2 + xy) |
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Answer» (i) f(x) = 21x2 y = 3 x 7x2 y g(x) = 35xy2 = 7 x 5xy2 G.C.D. = 7xy L.C.M. = 7 x 3 x 5 x x2 y2 = 105x2 x y2 L.C.M x G.C.D = f(x) x g(x) 105x2 y2 x 7xy = 21x2 y x 35xy2 735x3 y3 = 735x3 y3 Hence verified. (ii) (x3 – 1)(x + 1) = (x – 1)(x2 + x + 1)(x + 1) 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. x G.C.D = f(x) x g(x) (x – 1)(x + 1)(x2 + x + 1) (x2 – x + 1) = (x – 1) (x2 + x + 1) x (x + 1) (x2 – x + 1) (x3 – 1)(x + 1)(x3 + 1) = (x3 – 1)(x + 1)(x3 + 1) ∴ Hence verified. (iii) f(x) = x2 y + xy2 = xy(x + y) g(x) = x2 + xy = x(x + y) L.C.M. = x y (x + y) G.C.D. = x (x + y) To verify: L.C.M. x G.C.D. = xy(x + y) x (x + y) = x2 y (x + y)2 … (1) f(x) x g (x) = (x2 y + xy2)(x2 + xy) = x2 y (x + y)2 … (2) ∴ L.C.M. x G.C.D = f(x) x g{x). Hence verified. |
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