Question

If $\omega$ is the complex cube root of unity, then $(3 + 5\omega + 3\omega^2)^2 + (3 + 3\omega + 5\omega^2)^2 =$

• A. $-1$
• B. 0
• C. 4
• D. $-4$

Hint:

Whenever you see a quadratic trigonometric or polynomial structure with symmetric coefficients like $3 + 5\omega + 3\omega^2$, match the outer numbers first. Changing the coefficients of the outer terms to match the middle term using combinations of $1+\omega+\omega^2=0$ is almost always the fastest way to reduce the expression!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given an algebraic expression containing $\omega$, the complex cube root of unity. We need to simplify and find the numerical value of the expression $(3 + 5\omega + 3\omega^2)^2 + (3 + 3\omega + 5\omega^2)^2$.

Step 2: Key Formula or Approach:
We use the two fundamental properties of the complex cube roots of unity: $$1 + \omega + \omega^2 = 0 \implies 1 + \omega^2 = -\omega$$ $$\omega^3 = 1$$ We will regroup the terms inside the parentheses to form expressions that can be simplified using $3 + 3\omega^2 = 3(1+\omega^2) = -3\omega$.

Step 3: Detailed Explanation:
Let's analyze and simplify the first term: $$\text{Term 1} = 3 + 5\omega + 3\omega^2 = (3 + 3\omega^2) + 5\omega$$ Factor out 3: $$\text{Term 1} = 3(1 + \omega^2) + 5\omega$$ Substitute $1 + \omega^2 = -\omega$: $$\text{Term 1} = 3(-\omega) + 5\omega = -3\omega + 5\omega = 2\omega$$ Now, let's analyze and simplify the second term: $$\text{Term 2} = 3 + 3\omega + 5\omega^2 = 3(1 + \omega) + 5\omega^2$$ Substitute $1 + \omega = -\omega^2$: $$\text{Term 2} = 3(-\omega^2) + 5\omega^2 = -3\omega^2 + 5\omega^2 = 2\omega^2$$ Substitute these simplified terms back into our original expression: $$\text{Expression} = (2\omega)^2 + (2\omega^2)^2 = 4\omega^2 + 4\omega^4$$ Since $\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$: $$\text{Expression} = 4\omega^2 + 4\omega = 4(\omega^2 + \omega)$$ Using our identity property again, substitute $\omega^2 + \omega = -1$: $$\text{Expression} = 4(-1) = -4$$

Step 4: Final Answer:
The value of the expression is $-4$, which corresponds to option (D).