In a quadratic function, the value of the product of the roots (α, β) is 4.…
2016
In a quadratic function, the value of the product of the roots (α, β) is 4. Find the value of
\(\dfrac{\alpha^{n}+\beta^{n}}{\alpha^{-n}+\beta^{-n}}\)
- A.
\(n^4\) - B.
\(4^n\) - C.
\(2^{2n-1}\) - D.
\(4^{n-1}\)
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Correct answer: B
We are given that the product of the roots αβ = 4 and assume α and β are nonzero.
Compute the denominator:
α^{-n} + β^{-n} = 1/α^n + 1/β^n = (α^n + β^n)/(α^n β^n).
Now form the given ratio:
(α^n + β^n) / (α^{-n} + β^{-n}) = (α^n + β^n) / [ (α^n + β^n)/(α^n β^n) ] = α^n β^n.
Therefore the value equals (αβ)^n.
Since αβ = 4, the expression evaluates to 4^n.
Answer: 4^n
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