If α satisfies the equation x 2 + x + 1 = 0 and ( 1 + α ) 7 = A + B α + C α 2 , A , B , C ≥ 0 , then 5 ( 3 A - 2 B - C ) is equal to

Step-by-step solution

We know that, 1 + ω + ω 2 = 0 . So, the roots of the equation x 2 + x + 1 = 0 are ω and ω 2 . Let, α = ω ⇒ 1 + α 7 = 1 + ω 7 ⇒ 1 + α 7 = - ω 2 7 ⇒ 1 + α 7 = - ω 14 We know that, ω 3 n = 1 . ⇒ 1 + α 7 = - ω 2 = 1 + ω Now, on comparing with equation we get, ⇒ A = 1 , B = 1 , C = 0 ⇒ 5 3 A - 2 B - C = 5 3 - 2 - 0 ⇒ 5 3 A - 2 B - C = 5