Quartic Equations

x^4 - 8x^3 + 14x^2 - 8x + 1 = 0
x^2 - 8x + 14 - 8/x + 1/x^2 = 0
(x^2 + 2 + 1/x^2) - 8(x + 1/x) + 12 = 0
(x + 1/x)^2 - 8(x + 1/x) + 12 = 0
Let t = x + 1/x
t^2 - 8t + 12 = 0
(t - 2)(t - 6) = 0
t = 2 or t = 6
x + 1/x = 2 or x + 1/x = 6
x^2 - 2x + 1 = 0 or x^2 - 6x + 1 = 0
(x - 1)^2 = 0 or x = [6 +/- √(36 - 4)]/2
x = 1 or x = 3 +/- 2√2

By Factor Theorem, put x = 1, it satisfies the equation, so (x - 1) is a factor.
x^4 - 8x^3 + 14x^2 - 8x + 1 = (x - 1)(x^3 - 7x^2 + 7x - 1).
Again applying Factor Theorem to x^3 - 7x^2 + 7x - 1, (x- 1) is again a factor.
x^3 - 7x^2 + 7x - 1 = (x - 1)(x^2 - 6x + 1)
so x^4 - 8x^3 + 14x^2 - 8x + 1 = 0 becomes
(x - 1)^2 ( x^2 - 6x + 1) = 0
so the roots are 1, 1, (3 + 2sqrt 2) and (3 - 2sqrt 2).