Question

If \(z = 1 - \sqrt{3}\,i\), then \(z^3 - 3z^2 + 3z = \;?\)

• A. \(0\)
• B. \(1 + 3\sqrt{3}\,i\)
• C. \(1\)
• D. \(2 + 3\sqrt{3}\,i\)

Hint:

Many expressions like \(z^3 - 3z^2 + 3z\) can be written in factored form, often linked to \((z-1)^3\).
- Substituting specific complex numbers is simpler after factoring.

Answer 1

The Correct Option is B

Step 1: Recognize a binomial expansion pattern.
Notice \[ z^3 - 3z^2 + 3z \;=\; (z - 1)^3 + 1 \] because \[ (z-1)^3 = z^3 - 3z^2 + 3z - 1. \] Thus \[ z^3 - 3z^2 + 3z = (z - 1)^3 + 1. \] Step 2: Substitute \(z = 1 - \sqrt{3}\,i\).
Then \[ z - 1 \;=\; (1 - \sqrt{3}\,i) \;-\; 1 \;=\; -\sqrt{3}\,i. \] Hence \[ (z - 1)^3 \;=\; (-\sqrt{3}\,i)^3 = (-\sqrt{3})^3 \,(i^3) = -3\sqrt{3}\,\bigl(i^2 \cdot i\bigr) = -3\sqrt{3}\,\bigl(-1 \cdot i\bigr) = 3\sqrt{3}\,i. \] Therefore, \[ z^3 - 3z^2 + 3z = (z-1)^3 + 1 = 3\sqrt{3}\,i + 1 = \boxed{1 + 3\sqrt{3}\,i}. \]