Question

The Common roots of the equations z3 + 2z2 + 2z + 1 = 0 and z1985 + z100 + 1 = 0 are

Answer 1

Answer:

The given equation z3 + 2z2 + 2z + 1 = 0 can be rewritten as (z + 1) (z2 + z + 1) = 0. Its roots are

−1,  ω and ω2.

Let   f(z) = z1985 + z100 + 1

Putting z = −1, ω and ω2 respectively, we get

f(−1) = (−1)1985 + (−1)100 + 1 0

Therefore, −1 is not a root of the equation

f(z) = 0.

Again, f(w) = ω1985 + ω100+ 1

= (ω3)661 ω2 + (ω3)33 ω+1

= ω2 +ω + 1 = 0

Therefore, ω is a root of the equation f(z) = 0.

Similarly, f(ω2) = 0

Hence, ω and ω2 are the common roots.

Answer 2

Please refer to the answer above.