Demonstrate if 1x11x21x31x45 if x1x2x3x4 are roots of the eq

Demonstrate if (1-x1)(1-x2)(1-x3)(1-x4)=5 if x1,x2,x3,x4 are roots of the equation x^4 + x^3 + x^2 + x +1 =0

Solution

We could write Viete expressions:

x1 + x2 + x3 + x4 = -b/a=-1

x1*x2 + x1*x3 + x1*x4 + x2*x3 + x2*x4 + x3*x4=c/a=1

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 = -d/a=1

x1*x2*x3*x4= e/a=1

(1-x1)(1-x2)(1-x3)(1-x4)=1-(x1 + x2 + x3 + x4) + (x1*x2 + x1*x3 + x1*x4 + x2*x3 + x2*x4 + x3*x4) - (x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4) + (x1*x2*x3*x4)=

= 1+b/a+c/a+d/a+e/a=1+1+1+1+1=5 true