微積分證明一問

2010-02-28 5:07 am

Suppose that F,G and Q are polynomials and
F(x) / Q(x) = G(x) / Q(x)

for all x except when Q(x)=0, Prove that F(x) = G(x) for all x

煩請各位大大指教,感謝

回答 (2)

浮浪貢

2010-02-28 12:34 pm

✔ 最佳答案

設Q(x)=0有n個根:a1,a2,…,an
1.x >< ak(k=1~n),F(x)/ Q(x)=G(x)/ Q(x),=> F(x)=G(x)
2.x= ak(k=1~n), F(ak)=(x-> ak )lim F(x)……….F(x)連續函數
= (x-> ak )lim [F(x)* Q(x)]/ Q(x)
= (x-> ak )lim [G(x)* Q(x)]/ Q(x)
= (x-> ak )lim G(x)
= G(ak) ……….G(x)連續函數
由1.2.對所有的x, F(x)=G(x)

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天助

2010-03-01 10:45 pm

多項式F(x)-G(x)=0有無限多個根(總會超過deg(F(x)-G(x))),
so, F(x)-G(x)為零多項式, ie. F(x)=G(x) for all x.

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