f(x)=x^4+x^3+x^2+x+1 =
0.
x1,x2,x3 and x4 are the roots of the
equation.
To prove that (1+x1)(1+x2)(1+x3)(1+x4) =
1.
Since x1, x2 ,x3 and x4 are the roots of the equation,
f(x) = 0,
we can write the polynomial x^4+x^3+x^2+x+1 =
(x-x1)(x-x2)x-x3)(x-x4), by theory of equation, as the coefficient of x^4 agrees on both
sides.
Now put x =-1 on both
sides:
(-1)^4+(-1)^3+(1)^2+(-1)+1 =
(-1-x1)(1-x2)(-1-x3)(-1)-x4).
1-1+1-1+1 = (-1)^4
(1+x1)(1+x2)(1+x3)+(1+x4)
1 = (1+x1)(1+x2)(1+x3)(1+x4)
which is established.
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