Simplification of unity cube root

Question

If given that ω is a cube root of unity and using the properties ω, how can I simplify (1 + 2ω + 3ω²)(3 + 2ω + ω²)?

Answer 1

1) Multiple the two polynomials

2) Combine like terms

3) ω³ = 1, convert all

4) Combine like terms

5) ω⁴ = ω, convert all

6) Combine like terms

Answer 2

Hi there! So, the key here is to recognize that unity is a fancy way of saying the number 1. So, if ω is the cube root of unity,

ω³ = 1

We'll use this in a bit. First let's just take the expression they gave us and expand it with some distribution.

(1 + 2ω + 3ω²)(3 + 2ω + ω²)

1*3 + 1*2ω + 1*ω² + 2ω*3 + 2ω*2ω + 2ω*ω² + 3ω²*3 + 3ω²*2ω + 3ω²*ω²

3 + 2ω + ω² + 6ω + 4ω² + 2ω³ + 9ω² + 6ω³ + 3ω⁴

3 + 8ω + 14ω² + 8ω³ + 3ω⁴

Now we can substitute using the very first equation we wrote from the cube root fact.

3 + 8ω + 14ω² + 8(1) + 3(1)ω

3 + 8ω + 14ω² + 8 + 3ω

11 + 11ω + 14ω²

And there you have it, as simplified as we can get!