If z is a complex number, then the number of common roots of the equation z 1985 + z 100 + 1 = 0 and z 3 + 2 z 2 + 2 z + 1 = 0 , is equal to :

Step-by-step solution

Given, z 1985 + z 100 + 1 = 0 & z 3 + 2 z 2 + 2 z + 1 = 0 Now solving, z 3 + 2 z 2 + 2 z + 1 = 0 ⇒ z + 1 z 2 - z + 1 + 2 z z + 1 = 0 ⇒ z + 1 z 2 + z + 1 = 0 ⇒ z = - 1 , z = ω , ω 2 Now putting z = ω in z 1985 + z 100 + 1 we get, ⇒ ω 1985 + ω 100 + 1 ⇒ ω 2 + ω + 1 = 0 as ω 3 n = 1 Also, z = ω 2 ⇒ ω 3970 + ω 200 + 1 ⇒ ω + ω 2 + 1 = 0 Also, z = - 1 will not satisfy the equation z 1985 + z 100 + 1 Hence, there are two common root