(1) 因式分解 x^96 - 1 至最簡。請列式。
(2) 因式分解 x^108 - 1 至最簡。請列式。
數學知識交流 － 因式分解續
2012-06-22 5:54 am
回答 (3)
2012-06-22 6:41 am
✔ 最佳答案
1) x⁹⁶ - 1= (x⁴⁸ - 1)(x⁴⁸ + 1)= (x²⁴ - 1)(x²⁴ + 1)(x¹⁶ + 1)(x³² - x¹⁶ + 1)= (x¹² - 1)(x¹² + 1)(x⁸ + 1)(x¹⁶ - x⁸ + 1)(x¹⁶ + 1)(x³² - x¹⁶ + 1)= (x⁶ - 1)(x⁶ + 1)(x⁴ + 1)(x⁸ - x⁴ + 1)(x⁸ + 1)(x¹⁶ - x⁸ + 1)(x¹⁶ + 1)(x³² - x¹⁶ + 1)= (x³ - 1)(x³ + 1)(x² + 1)(x⁴ - x² + 1)(x⁴ + 1)(x⁸ - x⁴ + 1)(x⁸ + 1)(x¹⁶ - x⁸ + 1)= (x¹⁶ + 1)(x³² - x¹⁶ + 1)= (x - 1)(x² + x +1)(x + 1)(x² - x + 1) (x² + 1)(x⁴ - x² + 1)(x⁴ + 1)(x⁸ - x⁴ + 1)= (x⁸ + 1)(x¹⁶ - x⁸ + 1)(x¹⁶ + 1)(x³² - x¹⁶ + 1)= (x - 1)(x + 1)(x² + 1)(x² - x + 1)(x² + x +1)(x⁴ + 1)(x⁴ - x² + 1)(x⁸ + 1)= (x⁸ - x⁴ + 1)(x¹⁶ + 1)(x¹⁶ - x⁸ + 1)(x³² - x¹⁶ + 1)2) x¹ᴼ⁸ - 1= (x⁵⁴ - 1)(x⁵⁴ + 1)= (x²⁷ - 1)(x²⁷ + 1)(x¹⁸ +1)(x³⁶ - x¹⁸ + 1)= (x⁹ - 1)(x¹⁸ + x⁹ + 1)(x⁹ + 1)(x¹⁸ - x⁹ + 1)(x⁶ + 1)(x¹² - x⁶ + 1)(x³⁶ - x¹⁸ + 1)= (x³ - 1)(x⁶ + x³ + 1)(x¹⁸ + x⁹ + 1)(x³ +1)(x⁶ - x³ + 1)(x¹⁸ - x⁹ + 1)(x² + 1)= (x⁴ - x² + 1)(x¹² - x⁶ + 1)(x³⁶ - x¹⁸ + 1)= (x - 1)(x² + x + 1)(x⁶ + x³ + 1)(x¹⁸ + x⁹ + 1)(x + 1)(x² - x + 1)(x⁶ - x³ + 1)= (x¹⁸ - x⁹ + 1)(x² + 1)(x⁴ - x² + 1)(x¹² - x⁶ + 1)(x³⁶ - x¹⁸ + 1)= (x - 1)(x + 1)(x² + 1)(x² - x + 1)(x² + x + 1)(x⁴ - x² + 1)(x⁶ - x³ + 1)= (x⁶ + x³ + 1)(x¹² - x⁶ + 1)(x¹⁸ - x⁹ + 1)(x¹⁸ + x⁹ + 1)(x³⁶ - x¹⁸ + 1)
參考: My Maths World
2012-06-24 1:41 am
1.
x^96-1
=(x^48-1)(x^48+1)
=(x^24-1)(x^24+1)(x^48+1)
=(x^12-1)(x^12+1)(x^24+1)(x^48+1)
=(x^6-1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x^3-1)(x^3+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x-1)(x+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)(x^2+x+1)(x^2-x+1)
2.
x^108-1
=(x^54-1)(x^54+1)
=(x^27-1)(x^27+1)(x^54+1)
=(x^9-1)(x^18+x^9+1)(x^9+1)(x^18-x^9+1)(x^54+1)
=(x^3-1)(x^6+x^3+1)(x^18+x^9+1)(x^3+1)(x^6-x^3+1)(x^18-x^9+1)(x^54+1)
=(x^3-1)(x^3+1)(x^54+1)(x^6+x^3+1)(x^18+x^9+1)[x^6-(x^3-1)][x^18-(x^9-1)]
=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^54+1)(x^6+x^3+1)(x^18+x^9+1)[x^6-(x-1)(x^2+x+1)][x^18-(x^3-1)(x^6+x^3+1)]
x^96-1
=(x^48-1)(x^48+1)
=(x^24-1)(x^24+1)(x^48+1)
=(x^12-1)(x^12+1)(x^24+1)(x^48+1)
=(x^6-1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x^3-1)(x^3+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)
=(x-1)(x+1)(x^6+1)(x^12+1)(x^24+1)(x^48+1)(x^2+x+1)(x^2-x+1)
2.
x^108-1
=(x^54-1)(x^54+1)
=(x^27-1)(x^27+1)(x^54+1)
=(x^9-1)(x^18+x^9+1)(x^9+1)(x^18-x^9+1)(x^54+1)
=(x^3-1)(x^6+x^3+1)(x^18+x^9+1)(x^3+1)(x^6-x^3+1)(x^18-x^9+1)(x^54+1)
=(x^3-1)(x^3+1)(x^54+1)(x^6+x^3+1)(x^18+x^9+1)[x^6-(x^3-1)][x^18-(x^9-1)]
=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^54+1)(x^6+x^3+1)(x^18+x^9+1)[x^6-(x-1)(x^2+x+1)][x^18-(x^3-1)(x^6+x^3+1)]
2012-06-22 7:35 am
1. x^96 - 1
Solution: x^96 -1
= (x^48+1)(x^48-1)
= (x^48+1)(x^24+1)(x^24-1)
= (x^48+1)(x^24+1)(x^12+1)(x^12-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^6-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^3+1)(x^3-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^3+1)(x-1)(x^2+x+1)
2. x^108 - 1
Solution: x^108 -1
= (x^54+1)(x^54-1)
= (x^27+1)(x^27-1)(x^18+1)(x^36-x^18+1)
= (x^9-1)(x^18+x^9+1)(x^9+1)(x^18-x^9+1)(x^2+1)
= (x^3-1)(x^6+x^3+1)(x^18+x^9+1)(x^3+1)(x^6-x^3+1)(x^18-x^9-1)(x^2+1)
(x^4-x^2+1)(x^12-x^6+1)(x^36-x^18+1)
= (x-1)(x^2+x+1)(x^6+x^3+1)(x^18+x^9+1)(x+1)(x^2-x+1)(x^6-x^3+1)
(x^18-x^9+1)(x^2+1)(x^4-x^2+1)(x^12-x^6+1)(x^36-x^18+1)
= (x-1)(x+1)(x^2+1)(x^2-x+1)(x^2+x+1)(x^4-x^2+1)(x^6-x^3+1)
(x^6+x^3+1)(x^12-x^6+1)(x^18-x^9+1)(x^18+x^9+1)(x^36-x^18+1)
Solution: x^96 -1
= (x^48+1)(x^48-1)
= (x^48+1)(x^24+1)(x^24-1)
= (x^48+1)(x^24+1)(x^12+1)(x^12-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^6-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^3+1)(x^3-1)
= (x^48+1)(x^24+1)(x^12+1)(x^6+1)(x^3+1)(x-1)(x^2+x+1)
2. x^108 - 1
Solution: x^108 -1
= (x^54+1)(x^54-1)
= (x^27+1)(x^27-1)(x^18+1)(x^36-x^18+1)
= (x^9-1)(x^18+x^9+1)(x^9+1)(x^18-x^9+1)(x^2+1)
= (x^3-1)(x^6+x^3+1)(x^18+x^9+1)(x^3+1)(x^6-x^3+1)(x^18-x^9-1)(x^2+1)
(x^4-x^2+1)(x^12-x^6+1)(x^36-x^18+1)
= (x-1)(x^2+x+1)(x^6+x^3+1)(x^18+x^9+1)(x+1)(x^2-x+1)(x^6-x^3+1)
(x^18-x^9+1)(x^2+1)(x^4-x^2+1)(x^12-x^6+1)(x^36-x^18+1)
= (x-1)(x+1)(x^2+1)(x^2-x+1)(x^2+x+1)(x^4-x^2+1)(x^6-x^3+1)
(x^6+x^3+1)(x^12-x^6+1)(x^18-x^9+1)(x^18+x^9+1)(x^36-x^18+1)
收錄日期: 2021-04-13 18:45:46
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20120621000051KK00709