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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

一、k 重因式

二、重因式的判别和求法

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一、k重因式

定义9p(x)称为f (x)的k重因式,是指

p(x)不可约; pk(x)| f (x), 而 pk+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;

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.

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二、重因式的判别和求法

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

为 f(x) 的一阶微商(导数),f /(x)的微商(f /(x))/ = f //(x)为f(x) 的二阶微商等,f(x) 的k阶微商记为f (k)(x) .

  • 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 .
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例1.判别多项式   有无重因式.若有,求出其重因式.