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The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, is
Correct answer is '5'. Can you explain this answer?
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The least degree of a polynomial with integer coefficients, whose one...
Explanation:
Given:
One of the roots of the polynomial is cos 120.
Key Points:
- The roots of a polynomial with integer coefficients are algebraic numbers.
- The cosine of an angle is an algebraic number if the angle is a rational multiple of π.
Reasoning:
- The cosine of 120 degrees is -1/2.
- The polynomial with roots -1/2, α, β, γ, δ can be written as (x + 1/2)(x - α)(x - β)(x - γ)(x - δ) = 0.
- This polynomial has degree 5 because it has 5 linear factors.
Conclusion:
- The least degree of a polynomial with integer coefficients, whose one of the roots is cos 120, is 5.
Given:
One of the roots of the polynomial is cos 120.
Key Points:
- The roots of a polynomial with integer coefficients are algebraic numbers.
- The cosine of an angle is an algebraic number if the angle is a rational multiple of π.
Reasoning:
- The cosine of 120 degrees is -1/2.
- The polynomial with roots -1/2, α, β, γ, δ can be written as (x + 1/2)(x - α)(x - β)(x - γ)(x - δ) = 0.
- This polynomial has degree 5 because it has 5 linear factors.
Conclusion:
- The least degree of a polynomial with integer coefficients, whose one of the roots is cos 120, is 5.
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Community Answer
The least degree of a polynomial with integer coefficients, whose one...
If A = 12o, then 5A = 60o.
⇒ (4 cos3 A - 3 cos A)(2 cos2 A - 1) - (3 sin A - 4 sin3 A)(2 sin A cos A) = 1/2
⇒ (4 cos3 A - 3 cos A)(2 cos2 A - 1) - [3 - 4(1 - cos2 A)2(1 - cos2 A)cos A] = 1/2
On simplifying, this becomes a polynomial in cos A whose degree is 5.
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The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. Can you explain this answer?
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The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. 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 The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. Can you explain this answer?.
The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. 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 The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The least degree of a polynomial with integer coefficients, whose one of the roots be cos 120, isCorrect answer is '5'. Can you explain this answer?.
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