若f(x)=ax²+bx+c,满足f(x²+1)+f(x²)=2x的四次方+4.求f(x)表达式.

若f(x)=ax²+bx+c,满足f(x²+1)+f(x²)=2x的四次方+4.求f(x)表达式.
若f(x)=ax²+bx+c,满足f(x²+1)+f(x²)=2x的四次方+4.求f(x)表达式.

若f(x)=ax²+bx+c,满足f(x²+1)+f(x²)=2x的四次方+4.求f(x)表达式.
用配凑法比较简单.
f(x²+1)+f(x²)=2x⁴+4
=(x²+1)²+(x²)²-2x²-1+4
=(x²+1)²+(x²)²-(x²+1)-x² +2+2
=[(x²+1)²-(x²+1)+2]+[(x²)²-x²+2]
f(x²+1)=(x²+1)²-(x²+1)+2
f(x²)=(x²)²-x²+2
f(x)=x²-x+2
f(x)的表达式为f(x)=x²-x+2
此种方法可以避免繁琐的计算.

[a（x2+1）2+b(x2+1)+c] + [ax4+bx2+c] =2x4+4
2ax4+(2a+2b)x2+(a+b+2c)=2x4+4
同此项系数相同
a=1/2
2a+2b=0，b= -1/2
a+b+2c=4,c=2