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If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?
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If alpha beta gamma are the roots of the equation x cube Pnot X squa...
Solution:
Given equation: x^3 + Px^2 + Qx + R = 0
Let's assume that alpha, beta, and gamma are its roots.
Sum of roots:
α + β + γ = -P
Product of roots:
αβγ = -R
Now, let's find the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2)
1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = (1 - α^2) * (1 - β^2) * (1 - γ^2)
We can write (1 - α^2) as (1 + α) * (1 - α), and similarly for (1 - β^2) and (1 - γ^2).
So,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = (1 + α) * (1 - α) * (1 + β) * (1 - β) * (1 + γ) * (1 - γ)
= [(1 + α) * (1 + β) * (1 + γ)] * [(1 - α) * (1 - β) * (1 - γ)]
Now, let's find the value of (1 + α) * (1 + β) * (1 + γ) and (1 - α) * (1 - β) * (1 - γ)
(1 + α) * (1 + β) * (1 + γ) = 1 + (α + β + γ) + (αβ + βγ + γα) + αβγ
= 1 - P + Q - R
Similarly,
(1 - α) * (1 - β) * (1 - γ) = 1 - (α + β + γ) + (αβ + βγ + γα) - αβγ
= 1 + P + Q + R
Now,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = [(1 - α) * (1 - β) * (1 - γ)] * [(1 + α) * (1 + β) * (1 + γ)]
= (1 + P + Q + R) * (1 - P + Q - R)
= (1 - R^2)
Therefore, 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = 1 - R^2
Hence, the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) is equal to 1 - R^2.
Given equation: x^3 + Px^2 + Qx + R = 0
Let's assume that alpha, beta, and gamma are its roots.
Sum of roots:
α + β + γ = -P
Product of roots:
αβγ = -R
Now, let's find the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2)
1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = (1 - α^2) * (1 - β^2) * (1 - γ^2)
We can write (1 - α^2) as (1 + α) * (1 - α), and similarly for (1 - β^2) and (1 - γ^2).
So,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = (1 + α) * (1 - α) * (1 + β) * (1 - β) * (1 + γ) * (1 - γ)
= [(1 + α) * (1 + β) * (1 + γ)] * [(1 - α) * (1 - β) * (1 - γ)]
Now, let's find the value of (1 + α) * (1 + β) * (1 + γ) and (1 - α) * (1 - β) * (1 - γ)
(1 + α) * (1 + β) * (1 + γ) = 1 + (α + β + γ) + (αβ + βγ + γα) + αβγ
= 1 - P + Q - R
Similarly,
(1 - α) * (1 - β) * (1 - γ) = 1 - (α + β + γ) + (αβ + βγ + γα) - αβγ
= 1 + P + Q + R
Now,
(1 - α^2) * (1 - β^2) * (1 - γ^2) = [(1 - α) * (1 - β) * (1 - γ)] * [(1 + α) * (1 + β) * (1 + γ)]
= (1 + P + Q + R) * (1 - P + Q - R)
= (1 - R^2)
Therefore, 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) = 1 - R^2
Hence, the value of 1 - (α^2) * 1 - (β^2) * 1 - (γ^2) is equal to 1 - R^2.
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If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?
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If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?.
If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If alpha beta gamma are the roots of the equation x cube Pnot X square P1 X P2 equal to zero then find the value of 1 - minus Alpha square into 1 minus beta square into 1 minus gamma square?.
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