计算(X-1)(X-2)(X-3)(X-4)

来源：学生作业帮助网编辑：作业帮时间：2024/11/05 21:52:56

计算(X-1)(X-2)(X-3)(X-4)
计算(X-1)(X-2)(X-3)(X-4)

计算(X-1)(X-2)(X-3)(X-4)
(X-1)(X-2)(X-3)(X-4)
=(X-1)(X-4) (X-2)(X-3)
=（x^2-5x+4)(x^2-5x+6)
=(x^2-5x)^2+10(x^2-5x)+24
=x^4-10x^3+25x^2+10x^2-50x+24
=x^4-10x^3+35x^2-50x+24

=（x-1)（x-4)(x-2)(x-3)
=(x^2-5x+4)(x^2-5x+6）
设X^2-5为k
则（k+4)(k+6)=k^2+10K+24=(x^2-5X)^2+10(x^2-5x)+24=x^4-10x^3+25x^2+10x^2-50xd+24
=X^4-10x^3+35x^2-50x+24

let f(x)=
(x-1)(x-2)(x-3)(x-4) = x^4+ ax^3+bx^2+cx+d
f(0) = 24 = d
f(1) = 0 = 1+a+b+c +24 (1)
f(-1) = 120 = 1-a+b-c+24 (2)
(1) + (2)
120 =2b +50
...

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let f(x)=
(x-1)(x-2)(x-3)(x-4) = x^4+ ax^3+bx^2+cx+d
f(0) = 24 = d
f(1) = 0 = 1+a+b+c +24 (1)
f(-1) = 120 = 1-a+b-c+24 (2)
(1) + (2)
120 =2b +50
b = 35
f(2) = 0 = 16 + 8a + 4(35) + 2c +24
= 180 + 8a + 2c (3)
f(3) = 0 = 81 + 27a+ 9(35)+3c +24
= 420 + 27a + 3c (4)
2(4) - 3(3)
30a+300 =0
a = -10
c = -50

(x-1)(x-2)(x-3)(x-4) = x^4-10x^3+35x^2-50x+24

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