高中多項式餘數問題

Question

1.已知P（X）除以x-1餘數為2，除以x-3餘數為1.求p(x)除以(x-1)(x-2)的餘數。

2.x^100-2x^51+1 除以x^2-1的餘數

答案是次要，不過希望大家能幫我指出解題方法，謝謝！

Answer 1

最佳解答

1. 除以(x-1)(x-2)的餘數 à除以(x-1)(x-3)的餘數 P(1) = 2P(3) = 1 P(x) = (x-1)(x-3)Q(x) + (ax+b)P(1) = a + b = 2P(3) = 3a + b = 1 a = -1/2b = 5/2 p(x)除以(x-1)(x-3)的餘數為 –x/2 + 5/2 2.x^100-2x^51+1 = (x^2-1)Q(x) + (ax+b) 令x^2 = 1x^100 = (x^2)^50 = 1x^51 = (x)(x^50) = (x)((x^2)^25) = x ax + b ≡ 1- 2x + 1 = -2x + 2 餘數為 -2x + 2

Answer 2

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