本區搜索:
Yahoo!字典
打印

[問題] Factorization Problems (2)

[隱藏]

Factorization Problems (2)

2 problems this week.

1. (a) Factorize x^128 - x^64.
   (b) Hence, solve x^128 = x^64

2. Factorize 6x^5 + 6x^4 + 6

Try them all!
   

TOP

a) x^128 - x^64
= x^64 (x^64-1)
= x^64 (x^32+1)(x^32-1)
= x^64 (x^32+1)(x^16+1)(x^8+1)(x^4+1)(x^2+1)(x+1)(x-1)

b) x^128 = x^64
0= x^64 (x^32+1)(x^16+1)(x^8+1)(x^4+1)(x^2+1)(x+1)(x-1)
x=1 or -1 or 0

TOP

Q1a) x^128 - x^64
= x^64 (x^64 - 1)
= x^64 (x^32 + 1) (x^32 - 1)
= x^64 (x^32 + 1) (x^16 + 1) (x^16 - 1)
= x^64 (x^32 + 1) (x^16 + 1) (x^8 + 1) (x^8 - 1)
= x^64 (x^32 + 1) (x^16 + 1) (x^8 + 1) (x^4 + 1) (x^4 - 1)
= x^64 (x^32 + 1) (x^16 + 1) (x^8 + 1) (x^4 + 1) (x^2 + 1) (x^2 - 1)
= x^64 (x^32 + 1) (x^16 + 1) (x^8 + 1) (x^4 + 1) (x^2 + 1) (x + 1) (x - 1)

Q1b) x^128 = x^64
0 = x^128 - x^64
0 = x^64 (x^32 + 1) (x^16 + 1) (x^8 + 1) (x^4 + 1) (x^2 + 1) (x + 1) (x - 1)
x = 0  or  x + 1 = 0  or  x - 1 = 0
x = 0  or  x = -1  or  x = 1

Q2) 6x^5 + 6x^4 + 6
= 6 (x^5 + x^4 + 1)
= 6 [x^3 (x^2 + x + 1) - (x^3 - 1)]
= 6 [x^3 (x^2 + x + 1) - (x - 1) (x^2 + x + 1)]
= 6 {[x^3 - (x - 1)] (x^2 + x + 1)}
= 6 (x^3 - x + 1) (x^2 + x + 1)

TOP

重要聲明:小卒資訊論壇 是一個公開的學術交流及分享平台。 論壇內所有檔案及內容 都只可作學術交流之用,絕不能用商業用途。 所有會員均須對自己所發表的言論而引起的法律責任負責(包括上傳檔案或連結), 本壇並不擔保該等資料之準確性及可靠性,且概不會就因有關資料之任何不確或遺漏而引致之任何損失或 損害承擔任何責任(不論是否與侵權行為、訂立契約或其他方面有關 ) 。