Understanding the Problem: $\omega$ is a Cube Root of Unity

The question asks us to find the value of the expression $(1 - \omega + \omega^2) (1 + \omega - \omega^2)$ where $\omega$ is a cube root of unity. Problems involving $\omega$ is a cube root of unity often rely on understanding its specific properties.

Key Properties of $\omega$

When $\omega$ is a non-real cube root of unity, it satisfies the following fundamental properties:

• The sum of the cube roots of unity is zero: $1 + \omega + \omega^2 = 0$
• The cube of $\omega$ is one: $\omega^3 = 1$

These properties are crucial for simplifying expressions involving $\omega$. Since $\omega$ is a cube root of unity, we can use $1 + \omega + \omega^2 = 0$ to rearrange terms, for example, $1 + \omega = -\omega^2$, $1 + \omega^2 = -\omega$, and $\omega + \omega^2 = -1$.

Simplifying the Expression

Let's simplify the two factors in the given expression separately using the property $1 + \omega + \omega^2 = 0$.

Simplifying the First Factor: $(1 - \omega + \omega^2)$

We have the term $(1 - \omega + \omega^2)$. We can rearrange the property $1 + \omega + \omega^2 = 0$ to get $1 + \omega^2 = -\omega$.

Substitute this into the first factor:

$1 - \omega + \omega^2 = (1 + \omega^2) - \omega$

$1 - \omega + \omega^2 = (-\omega) - \omega$

$1 - \omega + \omega^2 = -2\omega$

Simplifying the Second Factor: $(1 + \omega - \omega^2)$

Now consider the second factor $(1 + \omega - \omega^2)$. From the property $1 + \omega + \omega^2 = 0$, we can get $1 + \omega = -\omega^2$.

Substitute this into the second factor:

$1 + \omega - \omega^2 = (1 + \omega) - \omega^2$

$1 + \omega - \omega^2 = (-\omega^2) - \omega^2$

$1 + \omega - \omega^2 = -2\omega^2$

Calculating the Final Value

Now we multiply the simplified factors:

$(1 - \omega + \omega^2) (1 + \omega - \omega^2) = (-2\omega) \times (-2\omega^2)$

$= (-2) \times (-2) \times \omega \times \omega^2$

$= 4 \times \omega^{1+2}$

$= 4 \times \omega^3$

We use the second key property that $\omega^3 = 1$ for any cube root of unity $\omega$.

So, $4 \times \omega^3 = 4 \times 1 = 4$.

Therefore, the value of the expression $(1 - \omega + \omega^2) (1 + \omega - \omega^2)$ when $\omega$ is a cube root of unity is 4.

Conclusion

By applying the basic properties of $\omega$ as a cube root of unity, namely $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$, we simplified the given expression step-by-step. Understanding these properties is essential for solving problems involving $\omega$ is a cube root of unity.