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MATH 問題

發問:

when P(x) is divided by x-1,the remainder is 1. when divided by (x-2)(x-3),the remainder is 5.find the remainder when P(x) is divided by (x-1)(x-2)(x-3) 更新: P(x)is a polynomial

• irrational numbers
• 商台o既獎
• Find a piece Aesop 's fables

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最佳解答:

Let P ( x ) = ( x - 2 )( x - 3 ) Q ( x ) + 5. P ( 2 ) = ( 2 - 2 )( 2 - 3 ) Q ( 2 ) + 5 = 5 P ( 3 ) = ( 3 - 2 )( 3 - 3 ) Q ( 3 ) + 5 = 5 So when P ( x ) is divided by x - 2 or x - 3, the remainder is 5. Next suppose P ( x ) = ( x - 1 )( x - 2 )( x - 3 )Q' ( x ) + px2 + qx + r. P ( 1 ) = p + q + r = 1 --- ( 1 ) P ( 2 ) = 4p + 2q + r = 5 --- ( 2 ) P ( 3 ) = 9p + 3q + r = 5 --- ( 3 ) From ( 1 ) and ( 2 ), 4p + 2q + 1 - p - q = 5 3p + q = 4 --- ( 4 ) From ( 2 ) and ( 3 ), 4p + 2q + r = 9p + 3q + r q = -5p --- ( 5 ) Put ( 5 ) into ( 4 ), 3p - 5p = 4 p = -2 q = ( - 5 )( - 2 ) = 10 Then, -2 + 10 + r = 1 r = -7 Hence the required remainder is -2x2 + 10x - 7.

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