If $x^2 - 8x + 1 = 0$, what is the value of $x^8 - 3842x^4 + 1$ ?

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We know that,

If x + \(\frac{1}{x}\)  = n

then, x² + \(\frac{1}{x^2}\)  = \(\sqrt {n^2 - 2}\)

If $x^2 - 8x + 1 = 0$,

what is the value of $x^8 - 3842x^4 + 1$ = ?

Here we can write $x^8 - 3842x^4 + 1$ as x⁴ + \(\frac{1}{x^4}\) - 3842

Divide on If $x^2 - 8x + 1 = 0$ both sides by x we get,

x + \(\frac{1}{x}\) = 8

x² + \(\frac{1}{x^2}\) = \(\sqrt {8^2 - 2}\) = 62

x⁴ + \(\frac{1}{x^4}\)= \(\sqrt {62^2 - 2}\) = 3842

Now put these above values in required equation,

x⁴ + \(\frac{1}{x^4}\) - 3842 = 3842 - 3842 = 0