Question

解方程6x^5-41x^4+97x^3-97x^2+41x-6=0

Answer 1

将原等式左边分解因式得如下过程：
6x^5-41x^4+97x^3-97x^2+41x-6
= (x-1)(6x^4-35x^3+62x^2-35x+6)
= (x-1)(x-2)(6x^3-23x^2+16x-3)
= (x-1)(x-2)(x-3)(6x^2-5x+1)
= (x-1)(x-2)(x-3)(2x-1)(3x-1) = 0
故得方程所有解：x=1,x=2,x=3,x=1/2,x=1/3,

Answer 2

6x^5 - 41x^4 + 97x^3 - 97x^2 + 41x - 6 = 0
6x^5 - 6 - 41x^4 + 41x + 97x^3 - 97x^2 = 0
6(x^5 - 1) - 41x(x³ - 1) + 97x²(x - 1) = 0
(x - 1)*(6x^4 - 35x³ + 6*x² - 35x + 6) = 0
…………
(x - 1)*(x - 2)*(x - 3)*(3x - 1)*(2x - 1) = 0
x = 1 x = 2 x = 3 x = 1/2 x = 1/3

Answer 3

6x^5-41x^4+97x^3-97x^2+41x-6
=(6x^5-6)-(41x^4-41x)+(97x^3-97x^2)
=6(x-1)(x^4+x^3+x^2+x+1)-41x(x-1)(x^2+x+1)+97x^2(x-1)
=(x-1)(6x^4-35x^3+62x^2-35x+6)
=(x-1)(3x^2-7x+2)(2x^2-7x+3)
=(x-1)(3x-1)(x-2)(2x-1)(x-3)
所以方程的解为1;1/3;2;1/2;3

Answer 4

6x^5-41x^4+97x^3-97x^2+41x-6=（3X-1）（2X-1）（X-1）（X-2）（X-3）=0
所以X=1/3 ， 1/2， 1， 2 或3