多项式f(x)以x-1除之余式为9,以x-2除之余式为16,求f(x)除以(x-1)(x-2)的余式.
问题描述:
多项式f(x)以x-1除之余式为9,以x-2除之余式为16,求f(x)除以(x-1)(x-2)的余式.
答
根据题意得:
∵f(x)=g(x)(x-1)+9①,
f(x)=h(x)(x-2)+16②,
∴②×(x-1)-①×(x-2)得:
[(x-1)-(x-2)]f(x)=[h(x)-g(x)](x-1)(x-2)+16(x-1)-9(x-2)
=[h(x)-g(x)](x-1)(x-2)+7x+2
∴f(x)除以(x-1)(x-2)的余式为7x+2.