請問:第10題該如何解呢？感恩~~~?

(x^15 + 1) (x + 1) / ( (x^5 + 1) (x³ + 1) )
= (x^5 + 1)(x^10 - x^5 + 1) (x + 1) / ( (x^5 + 1) (x³ + 1) )
= (x^10 - x^5 + 1) (x + 1) / (x³ + 1)

令 x³ = - 1 則 x^10 - x^5 + 1 = (-1)³x - (-1)x² + 1 = x² - x + 1,
故 (x^10 - x^5 + 1) (x + 1) = (x + 1) (Q(x) * (x³ + 1) + x² - x + 1) 被 x³ + 1 整除, 選項 (1) 錯誤.

= (x^10 - x^5 + 1) (x + 1) / ( (x + 1)(x² - x + 1) )
= (x^10 - x^5 + 1) / (x² - x + 1) ......... 選項 (2) 正確.

而 x⁴- x³ + x² - x + 1 不被 x² - x + 1 整除, 選項 (3) 錯誤.

f(x) = (Q(x) * (x³ + 1) + x² - x + 1) / (x² - x + 1)
= Q(x) * (x + 1) + 1 , 故 f(-1) = 1 , 選項 (3) , (5) 錯誤.

最後驗證 x^10 - x^5 + 1 = (x² - x + 1)(x^8 + x^7 - x^5 - x⁴- x³ + x + 1) , 選項 (4) 正確.