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# 一、 k 重因式 - PowerPoint PPT Presentation

1.6 重因式. 一、 k 重因式. 二、重因式的判别和求法. 一、 k 重因式. 定义 9 p ( x ) 称为 f ( x ) 的 k 重因式，是指 p ( x ) 不可约 ; p k ( x )| f ( x ), 而 p k+1 ( x ) 不能整除 f ( x ). 注：⑴ k = 0, p ( x ) 非 f ( x ) 的因式；. ⑵k = 1, p ( x ) 是 f ( x ) 的单因式；. ⑶ k ＞ 1, p ( x ) 是 f ( x ) 的重因式.

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1.6 重因式

p(x)不可约; pk(x)| f (x), 而 pk+1(x) 不能整除f (x).

⑵k = 1, p(x) 是 f (x) 的单因式；

⑶k＞1, p(x) 是 f (x) 的重因式.

⑷设f (x)典型分解为,则pi是f (x) 的 ri重因式(ri = 1,为单因式，ri＞1, 为重因式),i = 1,2,…,s.

1.定义 设 f(x) =anxn+ an－1xn－1 + ··· + a1x1 + a0 , 规定f /(x) = nanxn－1 + (n－1)an－1xn－2 + ··· + 2a2x + a1

• 简单性质:

1) (f + g)/ = f / + g /； (cf ) / = cf /；

(fg)/ = f /g + fg /； (fm(x))/ = m(f m－1(x)f /(x).

• 如上定义仅是一种形式的定义，并未赋予微商具体的，如同“数学分析”类似的意义. 又f (k)(x) ≠f k(x) .
• ∂f = n, 则有 ∂f / = n－1；∂ (f (n)) = 0；f(n+1) = 0.
• 例 f(x) = 3x4－5x3＋2x－1，f /(x) = 12x3－15x2＋2 .