✔ 最佳答案
<1>
deg f(x)=3,f(1)=f(2)=f(3)=4,f(4)=1,求f(x)=?
Sol
f(x)=a(x-1)(x-2)(x-3)+4
f(4)=a(4)(2)(1)+4=1
6a=-3
a=-1/2
f(x)=-(1/2)(x-1)(x-2)(x-3)+4
=-(1/2)(x^3-6x^2+11x-6)+4
=(-1/2)x^3+3x^2-(11/2)x+7
<2>degf(x)=4,f(x)除以(x-1)^3,餘式為3x-1,f(x)除以(x+2),餘74
f(x)除以(x-2),餘10,求f(x)=?
Sol
f(x)=a(x-1)^3(x+2)+b(x-1)^3+3x-1
f(-2)=b(-3)^3-7=74
b=-3
f(x)=a(x-1)^3(x+2)-3(x-1)^3+3x-1
f(2)=a(4)-3+5=10
a=2
f(x)=2(x-1)^3(x+2)-3(x-1)^3+3x-1
=2x^4-5x^3+3x^2+4x-2
<3>
a,b,c皆為整數,3x+2是(a+2)x^3+bx^2+cx+a的因式
又a介於200到300之間(含200,300),問此種a有多少個?
Sol
3|(a+2),2|a
a=2m
3|(2m+2)
200<=2m<=300
100<=m<=150
m=101,104,107,….,149
(149-101)/3+1=17
2012-02-03 00:50:08 補充:
degf(x)=4
f(x)除以(x-1)^3(x+2)設商為a,餘式q(x)
q(x)可能為3次,2次,1次,0次
設q(x)除以(x-1)^3商為b