设f(x)被x+1除的余式为10,被x+2除的余式为20,求f(x)被(x+1)(x+2)除的余式?

问题描述:

设f(x)被x+1除的余式为10,被x+2除的余式为20,求f(x)被(x+1)(x+2)除的余式?

f(x)=(x+1)(x+2)g(x)+ax+b (易知f(x)为2次多项)
f(-1)=10
f(-2)=20 (因式定理)
-a+b=10
-2a+b=20
a=-10
b=0
f(x)=(x+1)(x+2)g(x)-10x