Multiple choice

All the roots of the equation $11z^{10}+10iz^9+10iz-11=0$ lie

  1. Inside $|z|=1$
  2. On $|z|=1$
  3. Outside $|z|=1$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation is 11z^10 - 11 + 10iz^9 + 10iz = 0, which is 11(z^10 - 1) + 10iz(z^8 + 1) = 0. By Rouche's theorem or checking modulus, the roots lie on the unit circle |z|=1.

AI explanation

We can apply the property that if the coefficients of a polynomial satisfy a conjugate palindrome, meaning the k-th coefficient equals the conjugate of the (n minus k)-th coefficient, then all its roots lie on the unit circle. For the given equation 11 z to the power of 10 plus 10 i z to the power of 9 plus 10 i z minus 11 equals 0, the coefficients are 11 and minus 11, and 10 i and minus 10 i, which are conjugates with opposite signs. By the Cohn rule, dividing the polynomial by (z minus 1) preserves this reciprocal conjugate root property on the unit circle. Therefore, all roots of the given equation lie on the complex unit circle.

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