Re: [中學]幾題高中數學
※ 引述《bingo7 (夏天)》之銘言：
: 請教幾題高中數學,上次謝謝熱心的人
: 受益良多~
: 5.f(x)為三次多項式, f(x)除以x^2+x+2餘式為x+2, 除以x^2+x-2餘式為-x+2,
: 求f(x)除以x^3+x^2的餘式為?
f(x) = ax^3 + bx^2 + cx + d
= a(x + 1)x^2 + (b - a)x^2 + cx + d
= a[(x^2 + x - 2)x + 2x] + (b - a)x^2 + cx + d
= a(x^2 + x - 2)[ax + (b - a)] + (c + 3a - b)x + (d + 2b - 2a)
=> c + 3a - b = -1
d + 2b - 2a = 2
= a[(x^2 + x + 2)x - 2x] + (b - a)x^2 + cx + d
= (x^2 + x + 2)[ax + (b - a)] + (c - a - b)x + (d - 2b + 2a)
=> c - a - b = 1
d - 2b + 2a = 2
=> a = b = -1 / 2
d = 2
c = 0
所以餘式 = 2
: 7.設三次多項式f(x)被(x^2-1)除之得餘式為2x+3 , 以(x^2+1)除之得餘式為-2x-1 ,
: 則f(x)=?
f(x) = ax^3 + bx^2 + cx + d
= (x^2 - 1)[ax + b] + (c + a)x + (d + b)
=> c + a = 2
d + b = 3
= (x^2 + 1)[ax + b] + (c - a)x + (d - b)
=> c - a = -2
d - b = -1
=> c = 0
a = 2
d = 1
b = 2
=> f(x) = 2x^3 + 2x^2 + 1
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