If x^2- 5x + 1 = 0, then the value of ≤ft(x^4 + 1/(x^2)) ÷ (x^2 + 1)…

If x + \(\frac{1}{x}\)  = n

then, $x^3 +\frac{1}{x^3}$ = n³ - 3 × n

If $x^{2}- 5x + 1 = 0$

then the value of $\left(x^{4} + \frac{1}{x^{2}}\right) \div (x^{2} + 1)$ = ?

we can write $\left(x^{4} + \frac{1}{x^{2}}\right) \div (x^{2} + 1)$ by taking x as common form numerator and denominator as = 

$\left(x^{3} + \frac{1}{x^{3}}\right) \div (x + \frac{1}{x})$

$x^{2}- 5x + 1 = 0$

Divide by x on both sides,

x + \(\frac{1}{x}\) = 5

then, $x^3 +\frac{1}{x^3}$ = 5³ - 3 × 5 = 110

So the value of $\left(x^{3} + \frac{1}{x^{3}}\right) \div (x + \frac{1}{x})$ = \(\frac{110}{5}\) = 22