若(f(x),g(x))=1, 且 f(x)Ig(x)h(x),定理4则 f(x) I h(x).证: ： (f(x),g(x))=1,:. 3u(x),v(x)E P[x], 使u(x)f(x)+v(x)g(x)= 1于是有u(x)f(x)h(x) + v(x)g(x)h(x) = h(x)又 f(x)Ig(x)h(x), f(x)1 f(x)h(x):: f(x)Jh(x).RF81.4最大公因式§1.4 最大公因式 定理4 若 ，且 ， 则 ( f x g x ( ), ( ) 1 ) = f x g x h x ( ) | ( ) ( ) f x h x ( ) | ( ) . ( f x g x ( ), ( ) 1, ) =    u x v x P x ( ), ( ) [ ], 证： 使 u x f x v x g x ( ) ( ) ( ) ( ) 1 + = 于是有 u x f x h x v x g x h x h x ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = 又 f x g x h x ( ) | ( ) ( ), f x f x h x ( ) | ( ) ( )  f x h x ( ) | ( )