Question

(1-w+w^2)^2
Solution

Answer 1

Question:

Simplify ( 1 - ω + ω² )²; ω is cube root of unity.

Answer:

The simplified form of ( 1 - ω + ω² )² is - ( 4 + 4ω ).

Step-by-step-explanation:

We have given an expression.

We have to simplify the given expression.

The given expression is ( 1 - ω + ω² )².

We know that,

If ω is cube root of unity, ω² + ω + 1 = 0.

∴ ω² + ω = - 1

Now,

( 1 - ω + ω² )² = ( 1 - ω + ω² ) * ( 1 - ω + ω² )

⇒ ( 1 - ω + ω² )² = 1 ( 1 - ω + ω² ) - ω ( 1 - ω + ω² ) + ω² ( 1 - ω + ω² )

⇒ ( 1 - ω + ω² )² = 1 - ω + ω² - ω + ω² - ω³ + ω² - ω³ + ω⁴

⇒ ( 1 - ω + ω² )² = 1 - ω - ω + ω² + ω² + ω² - ω³ - ω³ + ω⁴

⇒ ( 1 - ω + ω² )² = 1 + ω + ω² - ω - ω - ω + 2ω² - 2ω³ + ω⁴

By using 1 + ω + ω² = 0, we get,

⇒ ( 1 - ω + ω² )² = 0 - 3ω + 2ω² - 2ω³ + ω⁴

⇒ ( 1 - ω + ω² )² = ω⁴ - 2ω³ + 2ω² - 3ω

⇒ ( 1 - ω + ω² )² = ω² ( ω² - 2ω + 2 ) - 3ω

⇒ ( 1 - ω + ω² )² = ω² ( ω² + ω + 1 - 3ω + 1 ) - 3ω

By using 1 + ω + ω² = 0, we get,

⇒ ( 1 - ω + ω² )² = ω² ( 0 - 3ω + 1 ) - 3ω

⇒ ( 1 - ω + ω² )² = ω² ( - 3ω + 1 ) - 3ω

⇒ ( 1 - ω + ω² )² = - 3ω³ + ω² - 3ω

By using ω³ = 1, we get,

⇒ ( 1 - ω + ω² )² = - 3 * 1 + ω² + ω - 4ω

⇒ ( 1 - ω + ω² )² = - 3 + ω² + ω - 4ω

By using ω² + ω = - 1, we get,

⇒ ( 1 - ω + ω² )² = - 3 - 1 - 4ω

⇒ ( 1 - ω + ω² )² = - 4 - 4ω

⇒ ( 1 - ω + ω² )² = - ( 4 + 4ω )

∴ The simplified form of ( 1 - ω + ω² )² is - ( 4 + 4ω ).