2x^4-3x^3+4x+5=a(x-1)^4+b(x-1)^3+c(x-1)^2+d(x-1)+e求a、b、c、d、e的值

a(x-1)^4+b(x-1)^3+c(x-1)^2+d(x-1)+e
= a(x^2 -2x + 1)^2 + b(x-1)(x^2 -2x + 1) + c(x^2 -2x + 1) + d(x - 1) + e
= (ax^4 - 4ax^3 + 6ax^2 - 4ax + a) + (bx^3 - 3bx^2 + 3bx -b) + (cx^2 - 2cx + c) + (dx - d) + e
= ax^4 + (b-4a)x^3 + (6a-3b+c)x^2 + (3b-4a-2c+d)x + (a-b+c-d+e)

对比系数可得

a = 2
b-4a = -3
6a-3b+c = 0
3b-4a-2c+d = 4
a-b+c-d+e = 5

解之，得
a = 2
b = 5
c = 3
d = 3
e = 8

X^4只有一组,故a=2
a[(x-1)^4]+b[(x-1)^3]+c[(x-1)^2]+d(x-1)+e
=2X^4+(b-8)*X^3+(12-3b+c)X^2+(2b-8-2c+d)X+2-b+c-d+e
所以
b-8=-3, b=5
12-3b+c=0,12-3*5+c=0,c=3
2b-8-2c+d=4,2*5-8-2*3+d=4,d=8
2-b+c-d+e=5,2-5+3-8+e=5,e=13

2x^4-3x^3+4x+5=a(x-1)^4+b(x-1)^3+c(x-1)^2+d(x-1)+e求a、b、c、d、e的值

1)首先判断：X^4只有一组,故a=2
2)令：x=1
2-3+4+5=e=====>e=8
3）令：x=2
2*16-3*8+4*2+5=2*1+b+c+d+8===>b+c+d=11.....(1)
4)x=0
5=2*1-b+c-d+8=====>b-c+d=5.....(2)
5)x=-1
2+3-4+5=2