若三次多项式g(x)的g(-1)=g(0)=g(2)=0g(1)=4问g(x)=?问若多项式h(x)=x^4-x^2+1则3g(x)-4h(x)被x-1除的余式

Question

若三次多项式g(x)的g(-1)=g(0)=g(2)=0 g(1)=4
问g(x)=?问若多项式h(x)=x^4-x^2+1则3g(x)-4h(x)被x-1除的余式

Answer 1

若三次多项式g(x)的g(-1)=g(0)=g(2)=0 ,g(1)=4 问g(x)=?
---
设g(x)=ax³+bx²+cx+d
g(-1)= -a+b-c+d=0
g(0)= d=0
g(2)=8a+4b+2c+d=0
g(1)= a+b+c+d=4
解得：
a=-2
b=2
c=4
d=0
g(x)=-2x³+2x²+4x
----------------------------------------------------
若多项式h(x)=x^4-x^2+1 则3g(x)-4h(x)被x-1除的余式
---
3g(x)-4h(x)
=-6x³+6x²+12x-(4x^4-4x^2+4)
=-4x^4-6x³+10x^2+12x-4
=-2x²(2x²+3x-5)+12x-4
=-2x²(x-1)(2x+5)+12(x-1)+8
3g(x)-4h(x)被x-1除的余式是8