If y = x + 1/x , then x^4 + x^3 – 4x^2 + x + 1 = 0 can be reduced to which one of the following ? (x ≠ 0)

Question

If y = \(x+\frac{1}{x}\) then x⁴ + x³ – 4x² + x + 1 = 0 can be reduced to which one of the following ? (x ≠ 0)

(a) y² + y – 2 = 0 

(b) y² + y – 4 = 0 

(c) y² + y – 6 = 0 

(d) y² + y + 6 = 0

Answer 1

(c) y² + y – 6 = 0

x⁴ + x³ – 4x² + x + 1 = 0

⇒ x² + x - 4 + \(\frac{1}{x}\) + \(\frac{1}{x^2}\) = 0

⇒ \(x^2+\frac{1}{x^2}\) + \(x+\frac{1}{x}\) - 4 = 0 

⇒ \(\big(x+\frac{1}{x}\big)^2\) - 2 + \(\big(x+\frac{1}{x}\big)\) - 4 = 0

⇒ y² + y – 6 = 0                      \(\big(∵x+\frac{1}{x}=y\big)\)