Question

If the cube roots of the unity are 1, ω and ω², then the roots of the equation (x – 1)³ + 8 = 0, are ______.

Options

• –1, 1 + 2ω, 1 + 2ω²

• –1, 1 – 2ω, 1 – 2ω²

• –1, –1, –1

• –1, –1 + 2ω, – 1 – 2ω²

Answer 1

If the cube roots of the unity are 1, ω and ω², then the roots of the equation (x – 1)³ + 8 = 0, are –1, 1 – 2ω, 1 – 2ω².

Explanation:

Given that, (x – 1)³ + 8 = 0

`\implies` (x – 1)³ = (–2)³

`\implies ((x - 1)/-2)^3` = 1

`\implies ((x - 1)/-2)` = (1)^1/3

∴ Cube roots of `((x - 1)/-2)` are 1, ω and ω².

Cube roots of (x – 1) are –2, –2ω arid –2ω².

Cube roots of x are –1, 1 – 2ω and 1 – 2ω².