Re: [中學] 代數一題
※ 引述《ghkckhg (台灣自耕農代表)》之銘言:
: 題目如下:
: 第12題
: http://ppt.cc/,phg
: 看了許久看不出端倪...
: 莫非真是什麼羚羊掛角之招?
: 謝謝~
很不幸的,這題是有trick的
這裡三次方是嚇人的, 可以不管
考慮函數 f(T):=x/(T-1)+y/(T-3^3)+z/(T-5^3)+w/(T-7^3) -1
Then f(T)=0 for T=2^3,4^3,6^3,8^3.
(T-1)(T-3^3)(T-5^3)(T-7^3)f(T) 是一個T的四次多項式且有零點 2^3,4^3,6^3,8^3
顯然 (T-1)(T-3^3)(T-5^3)(T-7^3)f(T) = -(T-2^3)(T-4^3)(T-6^3)(T-8^3).
比較三次項係數, x+y+z+w+1+3^3+5^3+7^3 = 2^3+4^3+6^3+8^3.
so, x+y+z+w = 2^3+4^3+6^3+8^3 -(1+3^3+5^3+7^3).
題外話, 多算幾步還能解出x,y,z,w:
-(T-2^3)(T-4^3)(T-6^3)(T-8^3)
In fact, f(T)= --------------------------------
(T-1)(T-3^3)(T-5^3)(T-7^3)
5383385 9205785 35906247 222893255
=-1 + ----------- + ------------- + ------------- + ------------
122512(T-1) 115024(T-3^3) 378448(T-5^3) 2617744(T-7^3)
--
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