Question

find the LCMand GCD for the following and verify that f(x) × g(x) = LCM×GCD
1. 21x² y ,35 xy²
2. (x³ _1) (x+1) , (x³+1)
3. ( x²y+xy²) ,(x²+xy) ​

Answer 1

Step-by-step explanation:

(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 y  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.