設f(x)=4x2↑ + 14x + 12 及 g(x) = 2x3↑+ 9 x2↑ + 13x + 6 (a) 求 f(x)和g(x)的H.C.F = 和L.C.M (b) 驗證 f(x) X g(x) = H.C.F X L.C.M 謝謝!謝謝!?

f(x)=4x^2 +14x+12
f(x)=2(2x+3)(x+2)
g(x)=2x^3 +9x^2+13x+6
g(x)=(x+1)(2x+3)(x+2)

a)
HCF=(2x+3)(x+2)
LCF=2(x+1)(2x+3)(x+2)

b)
f(x) X g(x)
=2(2x+3)(x+2)(x+1)(2x+3)(x+2)
=2(2x+3)^2 (x+2)^2 (x+1)

H.C.F X L.C.M
=2(x+1)(2x+3)(x+2)(2x+3)(x+2)
=2(2x+3)^2 (x+2)^2 (x+1)
=f(x) X g(x)

(a)
f(x) = 4x² + 14x + 12
= 2(2x² + 7x + 6)
= 2(x + 2)(2x + 3)

g(x) = 2x³ + 9x² + 13x + 6
= 2x³ + 2x² + 7x² + 7x + 6x + 6
= 2x²(x + 1) + 7x(x + 1) + 6(x + 1)
= (x + 1)(2x² + 7x + 6)
= (x + 1)(x + 2)(2x + 3)

H.C.F. = (x + 2)(2x + 3) = 2x² + 7x + 6
L.C.M. = 2(x + 1)(x + 2)(2x + 3) = 2g(x) = 4x³ + 18x² + 26x + 12

(b)
f(x) × g(x)
= (4x² + 14x + 12)(2x³ + 9x² + 13x + 6)
= 2(x + 2)(2x + 3)(x + 1)(x + 2)(2x + 3)
= 2(x + 1)(x + 2)²(2x + 3)²

H.C.F. × L.C.M.
= (2x² + 7x + 6)(4x³ + 18x² + 26x + 12)
= (x + 2)(2x + 3)2(x + 1)(x + 2)(2x + 3)
= 2(x + 1)(x + 2)²(2x + 3)²
= f(x) × g(x)