If the highest common factor of f(x) and f'(x) contains a k-multiple factor (x - a), then a is a (k + 1) - multiple root of equation f(x) = 0 ( Let f(x) be a polynomial of degree n >=1)
Why??
更新1:
Q(x) = [kP(x)+(x-a)P'(x)]??
A question (2)
2009-12-24 1:50 am
Steps must shown clearly.
If the highest common factor of f(x) and f'(x) contains a k-multiple factor (x - a), then a is a (k + 1) - multiple root of equation f(x) = 0 ( Let f(x) be a polynomial of degree n >=1)
Why??
If the highest common factor of f(x) and f'(x) contains a k-multiple factor (x - a), then a is a (k + 1) - multiple root of equation f(x) = 0 ( Let f(x) be a polynomial of degree n >=1)
Why??
回答 (1)
2009-12-24 3:25 am
✔ 最佳答案
We have f(x)=(x-a)^k P(x) ; f'(x)=(x-a)^k Q(x)On the other hand f'(x)=k(x-a)^(k-1) P(x) + (x-a)^k P'(x)
=(x-a)^(k-1) [kP(x)+(x-a)P'(x)]
Compare with x-a)^k Q(x) => P(x) has a factor (x-a)
So f(x)=(x-a)^(k+1) R(x) where P(x)=(x-a)R(x) and a is a (k + 1) - multiple root of equation f(x) = 0
2009-12-26 13:24:07 補充：
(x-a)^k Q(x) = (x-a)^(k-1) [kP(x)+(x-a)P'(x)]
收錄日期: 2021-04-26 13:50:00
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https://hk.answers.yahoo.com/question/index?qid=20091223000051KK01324