Question

If \( z = e^{2\pi i / 3} \), then \( 1 + z + 3z^2 + 2z^3 + 2z^4 + 3z^5 \) is equal to:

• A. \( -3e^{\pi i / 3} \)
• B. \( 3e^{\pi i / 3} \)
• C. \( 3e^{2\pi i / 3} \)
• D. \( -3e^{2\pi i / 3} \)
• E. \( 0 \)

Hint:

When you see $e^{2\pi i / n}$, you are working with $n$-th roots of unity. Always group the terms in sets of $n$ to exploit the fact that their sum is zero.

Answer 1

The Correct Option is A

Concept: The value \( z = e^{2\pi i / 3} \) is one of the complex cube roots of unity, commonly denoted as \( \omega \). Properties of \( \omega \):
• \( \omega^3 = 1 \)
• \( 1 + \omega + \omega^2 = 0 \) We simplify the given expression using these properties.

Step 1: Simplify higher powers of \( z \).
Using \( z^3 = 1 \): \[ z^3 = 1 \] \[ z^4 = z^3 \cdot z = z \] \[ z^5 = z^3 \cdot z^2 = z^2 \]

Step 2: Substitute back into the expression.
\[ 1 + z + 3z^2 + 2(1) + 2(z) + 3(z^2) \] Combine like terms: \[ (1 + 2) + (z + 2z) + (3z^2 + 3z^2) \] \[ 3 + 3z + 6z^2 \] \[ 3(1 + z + 2z^2) \]

Step 3: Apply \( 1 + z + z^2 = 0 \).
Since \( z^2 = -(1 + z) \): \[ 3(1 + z + 2z^2) = 3(1 + z + z^2 + z^2) = 3(0 + z^2) = 3z^2 \]

Step 4: Express in exponential form.
\( 3z^2 = 3(e^{2\pi i / 3})^2 = 3e^{4\pi i / 3} \). Note that \( e^{4\pi i / 3} = e^{4\pi i / 3 - 2\pi} = e^{-2\pi i / 3} \). Also, \( e^{4\pi i / 3} = e^{\pi i} \cdot e^{\pi i / 3} = -1 \cdot e^{\pi i / 3} \). \[ 3z^2 = -3e^{\pi i / 3} \]