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第一次习题è®&um

第一次习题è®&um

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1 Exercise 1 a, b, c x - a, x - b, x - c 除၂ჭ多项 f ( x ) 的Ⴥ式၇次为 r, s, t ,试求Ⴈ g ( x ) = ( x - a )( x - b )( x - c ) f ( x ) Ⴥ式。 解: 设 f ( x ) = q ( x ) g ( x ) + r ( x ) ,ᄵ可设 r ( x ) = a 2 x 2 + a 1 x + a 0 Ⴎ Ⴟ g ( x ) 是 三 次 多 项 式 , 故, r ( x ) 不超过二次。 r ( a ) = f ( a ) = r r ( b ) = f ( b ) = s r ( c ) = f ( c ) = t ႮႿ g ( a ) = g ( b ) = g ( c ) = 0 故Ⴕ如下方程ቆ: 1 a a 2 1 b b 2 1 c c 2 a 0 a 1 a 2 = r s t 容ၞ通过ᆃ个方程ቆ解出 a 0 , a 1 , a 2 的ᆴ,从而得到 r ( x ) Exercise 2 p, q, r ᆭ间的关系,使得 x 3 + px 2 + qx + r 的根成等比 数列。 解: 设 x 3 + px 2 + qx + r 的三个根分别为 x 0 a , x 0 , ax 0 ,ᄵႵ x 3 + px 2 + qx + r = ( x - x 0 a )( x - x 0 )( x - ax 0 ) 将Ⴗ式ᅚ开,便Ⴕ: p = - ( x 0 a + x 0 + ax 0 ) q = x 2 0 a + x 2 0 + ax 2 0 r = - x 3 0 Ⴎ此,便Ⴕ ( q p ) 3 = r Exercise 3 如果任ၩ多项式或ᆀა多项式 p ( x ) 互素,或ᆀ能被 p ( x ) 除,试ᆣ明 p ( x ) 不可ჿ。 ᆣ明: Ⴈ反ᆣ法。 假设 p ( x ) 可ჿ,ᄵ不妨设 p ( x ) = p 1 ( x ) p 2 ( x ) ,其ᇏ, deg p i ( x ) < deg p ( x ) , i = 1 , 2 。从而 p 1 ( x ) 既不ა p ( x ) 互素,Ⴛ不能被 p ( x ) ᆜ除。矛盾! 故, p ( x ) 不可ჿ。ᆣ毕。 Exercise 4 集合 R n = { ( a 1 , a 2 , . . . , a n ) T | a i R } 关Ⴟ矩ᆔ的加法和数 (1) 是否构成Ⴕ理数თ上的线性空间? (2) 是否构成复数თ上的线性空间? 解: (1) 显然, R n 关Ⴟ矩ᆔ的加法和数乘,构成Ⴕ理数თ上的线性空 ႮႿ R n 对Ⴟ实数თ构成线性空 间,而Ⴕ理数თ包含Ⴟ实数თ, ၹ此其对ႿႵ理数თ必构成线性 空间。 间。 1
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(2) R n
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