Re: [中學] 多項式
※ 引述《wanling0419 (Wanling)》之銘言：
: 在各次係數皆為1或2或3的多項式函數f(x)中，
: 有____個滿足f(2)=101
: 想請問除了一一列出討論還有其他方式嗎？
( x^6 + x^5 + ... + x + 1 ) | x=2
= 127
3 ( x^4 + ... + x + 1) | x=2
= 93
deg( f(x) ) = 5
=> 32 a + 16 b + 8 c + 4 d + 2 e + f = 101
32 a + 16 b + 8 c + 4 d + 2 e + f | a=b=c=d=e=f=1
= 63
=> 32 a + 16 b + 8 c + 4 d + 2 e + f = 38
(0<=a,b,c,d,e,f<=2)
if f = 0
=> 16 a + 8 b + 4 c + 2 d + e = 19
if a = 1 => b = c = 0 => (d,e)=(1,1) 1種
if a = 0 => (b,c,d,e) = (2,0,1,1) (1,2,1,1) 2種
if f = 2
=> 16 a + 8 b + 4 c + 2 d + e = 18
if a = 1 => b = c = 0 => (d,e) = (0,2) (1,0) 2種
if a = 0 => (b,c,d,e) = (2,0,0,2) (2,0,1,0) (1,2,1,0) (1,2,0,2) (1,1,2,2) 5種
=> 10
希望沒有漏算XD
有錯還請不吝指正
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