If x + (1/x) = p, then x^6+(1/x^6) equals to :

Question

If \(x+\frac{1}{x}\) = p, then \(x^6+\frac{1}{x^6}\) equals to :

(a) p⁶ + 6p 

(b) p⁶ – 6p 

(c) p⁶ + 6p⁴ + 9p² + 2 

(d) p⁶ – 6p⁴ + 9p² – 2

Answer 1

(d) p⁶ – 6p⁴ + 9p² – 2

Given, \(x+\frac{1}{x}\) = p          ⇒ \(\bigg(x+\frac{1}{x}\bigg)^2\) = p²

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

⇒ \(\bigg(x^2+\frac{1}{x^2}\bigg)^3\) = (p² - 2)³

⇒  \(x^6+\frac{1}{x^6}\) + 3\(\bigg(x^2+\frac{1}{x^2}\bigg)\) = p⁶ - 8 + 6p² (p² - 2)

⇒ \(x^6+\frac{1}{x^6}\) + 3 (p² - 2) = p⁶ - 8 + 6p² (p² - 2)

⇒ \(x^6+\frac{1}{x^6}\) = p⁶ - 6p⁴ - 9p² - 2