यदि x² + x + 1 = 0, तो \(\left(x+\dfrac{1}{x}\right)^2+\left(x^2+\dfrac{1}{x^2}\right)^2+\left(x^3+\dfrac{1}{x^3}\right)^2+.....\left(x^{21}+\dfrac{1}{x^{21}}\right)^2\) का मान ज्ञात करें।

x² + x + 1 = 0

\({\left( {x+\frac{1}{x}} \right)^2} + {\left( {{x^2}+\frac{1}{{{x^2}}}} \right)^2}\)+\({\left( {{x^3} + \frac{1}{{x3}}} \right)^2} + ... + {\left( {{x^{21}} + \frac{1}{{{x^{21}}}}} \right)^2}\)

जैसा कि हम जानते हैं

x² + x + 1 = 0

x³ + x² + x = 0

x³ + x² = -x

x³ + x² = 1 + x²

x³ = 1

x² + x + 1 = 0

  \(x + 1 + \frac{1}{x}=0\)

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

\(x^2+\frac{1}{x^2}+2\times x\times\frac{1}{x}=1\)

\(x^2+\frac{1}{x^2}=-1\)

इसी तरह,