Question

把多项式x三次方-x方+2x+2表示成a(x-1)三次方+b(x-1)方+c(x-1)+d的形式

Answer 1

解 根据题意有：x^3 - x^2 + 2x + 2 = a(x-1)^3 + b(x-1)^2 + c(x-1) + d
由于 a(x-1)^3 + b(x-1)^2 + c(x-1) + d
=a(x^3 - 3x^2 + 3x - 1) + b(x^2 - 2x + 1) + cx - c + d
= ax^3 - 3ax^2 + 3ax - a + bx^2 - 2bx + b + cx - c + d
= ax^3 - (3a -2b)x^2 + (3a - 2b + c)x + (-a + b - c + d)
所以有： x^3 - x^2 + 2x + 2 = ax^3 - (3a - 2b)x^2 + (3a -2b + c)x + (-a + b - c + d)
比较等式两边系数有： a = 1 (1)
3a - 2b = 1 (2)
3a - 2b + c = 2 (3)
-a + b - c + d = 2 (4)
联立（1）（2）（3）（4）解得： a = 1 b = 1 c = 1 d = 3
所以： x^3 - x^2 + 2x + 2 = (x - 1)^3 + (x - 1)^2 + (x - 1) + 3