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If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)
- a)– 1, 1 + 2ω, 1 + 2ω2
- b)– 1, 1 – 2ω, 1 – 2ω2
- c)– 1, – 1, – 1
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If the cube roots of unity are 1, ω, ω2, then the roots of...
(x – 1)3 + 8 = 0
⇒ (x – 1)3 = – 8 = (– 2)3
⇒ x – 1 = – 2
⇒ (x – 1)3 = – 8 = (– 2)3
⇒ x – 1 = – 2
or -2ω or -2ω2
⇒ x = -1, 1 - 2ω, 1 -2ω2
Most Upvoted Answer
If the cube roots of unity are 1, ω, ω2, then the roots of...
$e^{\frac{2\pi i}{3}}$, and $e^{\frac{4\pi i}{3}}$, then we can use these values to find the roots of a cubic polynomial.
Let's say we want to find the roots of the polynomial $x^3 + 2x^2 + x - 2$. We can factor it as $(x-1)(x^2+3x+2)$, so one root is $x=1$.
To find the other two roots, we can use the quadratic formula on $x^2+3x+2$. We get:
$$x = \frac{-3 \pm \sqrt{9-8}}{2} = -1,\,-2$$
Now we can use the cube roots of unity to express these roots in terms of complex numbers.
For $x=-1$, we have:
$$x = -1 = -1 \cdot 1 = -1 \cdot e^{0i}$$
For $x=-2$, we have:
$$x = -2 = -2 \cdot 1 = -2 \cdot e^{0i}$$
So the roots of the polynomial can be expressed as:
$$x = 1,\,-1,\,-2$$
These are the three cube roots of unity, as expected.
Let's say we want to find the roots of the polynomial $x^3 + 2x^2 + x - 2$. We can factor it as $(x-1)(x^2+3x+2)$, so one root is $x=1$.
To find the other two roots, we can use the quadratic formula on $x^2+3x+2$. We get:
$$x = \frac{-3 \pm \sqrt{9-8}}{2} = -1,\,-2$$
Now we can use the cube roots of unity to express these roots in terms of complex numbers.
For $x=-1$, we have:
$$x = -1 = -1 \cdot 1 = -1 \cdot e^{0i}$$
For $x=-2$, we have:
$$x = -2 = -2 \cdot 1 = -2 \cdot e^{0i}$$
So the roots of the polynomial can be expressed as:
$$x = 1,\,-1,\,-2$$
These are the three cube roots of unity, as expected.
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If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the cube roots of unity are 1, ω, ω2, then the roots of the equation (x – 1)3 + 8 = 0 are (1979)a)– 1, 1 + 2ω, 1 + 2ω2b)– 1, 1 – 2ω, 1 – 2ω2c)– 1, – 1, – 1d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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