Complex Numbers
The cube roots of unity are
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Solution
1, ω, ω²
- Equation: Cube roots of unity satisfy x3=1.
- Factorization: x3−1=0⟹(x−1)(x2+x+1)=0.
- Solve Quadratic: x2+x+1=0⟹x=2−1±1−4=2−1±i3.
- Designations:
- ω=2−1+i3
- ω2=2−1−i3
- Properties: Sum of roots 1+ω+ω2=0 and product ω3=1.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- complex numbers
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
1, ω, ω²
- Equation: Cube roots of unity satisfy x3=1.
- Factorization: x3−1=0⟹(x−1)(x2+x+1)=0.
- Solve Quadratic: x2+x+1=0⟹x=2−1±1−4=2−1±i3.
- Designations:
- ω=2−1+i3
- ω2=2−1−i3
- Properties: Sum of roots 1+ω+ω2=0 and product ω3=1.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of complex numbers. It appeared in the 2025 exam.
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