Question

If
\[ iz^3+z^2-z+i=0, \] then \(|z|=\)

• A. \(\dfrac{1}{2}\)
• B. \(2\)
• C. \(\dfrac{3}{2}\)
• D. \(1\)

Hint:

In complex number problems, try factorization first. Also remember that if \(z^2=w\), then \(|z|^2=|w|\).

Answer 1

The Correct Option is D

Step 1: Write the given equation.
We have
\[ iz^3+z^2-z+i=0 \]
Group the terms:
\[ (iz^3+z^2)+(-z+i)=0 \]
Factor each group:
\[ z^2(iz+1)-1(z-i)=0 \]
Observe that
\[ z-i=-i(iz+1) \]
Hence,
\[ z^2(iz+1)+i(iz+1)=0 \]
Taking common factor \((iz+1)\), we get
\[ (iz+1)(z^2+i)=0 \]

Step 2: Find the roots.
Case 1:
\[ iz+1=0 \] \[ iz=-1 \] \[ z=\frac{-1}{i} \] \[ z=i \]
Thus,
\[ |z|=|i|=1 \]
Case 2:
\[ z^2+i=0 \] \[ z^2=-i \]
Now,
\[ |-i|=1 \]
Taking modulus on both sides,
\[ |z|^2=1 \] \[ |z|=1 \]
Thus, every root satisfies
\[ |z|=1 \]

Step 3: Final conclusion.
Hence,
\[ \boxed{1} \]