The HCF of x⁶ - 1 and x⁴ + 2x³ - 2x¹ - 1 is = ?

Solution(By Examveda Team)

$$\eqalign{ & {x^6} - 1 \Rightarrow {\left( {{x^2}} \right)^3} - {1^3} \cr & {\text{Using}} \cr & \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right) \cr & \Rightarrow \left( {{x^2} - 1} \right)\left( {{x^4} + 1 + {x^2} \times 1} \right) \cr & \Rightarrow \left( {{x^2} - 1} \right)\left( {{x^4} + 1 + {x^2}} \right).....(i) \cr & {\text{Again}} \cr & {x^4} + 2{x^3} - 2x - 1 \cr & \Rightarrow {x^4} - 1 + 2x\left( {{x^2} - 1} \right) \cr & \Rightarrow {\left( {{x^2}} \right)^2} - {1^2} + 2x\left( {{x^2} - 1} \right) \cr & \Rightarrow \left( {{x^2} - 1} \right)\left( {{x^2} + 1} \right) + 2x\left( {{x^2} - 1} \right) \cr & \Rightarrow \left( {{x^2} - 1} \right)\left( {{x^2} + 1 + 2x} \right).....(ii) \cr} $$
  ⇒ From equation (i) and (ii) HCF is a common term HCF = (x² - 1)