Question

关于x的方程x^4+(k-1)x^3+kx^2+(k-1)x+1=0没有实数根,求k的取值范围.

Answer 1

x^4+(k-1)x^3+kx^2+(k-1)x+1=0

↔ x^4-x^3-x+1+kx^3+kx^2+kx=0

↔ x^3*(x-1)-(x-1)+kx*(x^2+x+1)=0

↔ (x^3-1)(x-1)+kx*(x^2+x+1)=0

↔ (x^2+x+1)*(x-1)^2+kx*(x^2+x+1)=0

↔ (x^2+x+1)*[(x-1)^2+kx]=0

要使方程x^4+(k-1)x^3+kx^2+(k-1)x+1=0没有实数根,则要求方程(x^2+x+1)*[(x-1)^2+kx]=0

而方程x^2+x+1=0的△=(-1)^2-4=-3