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irudayadhason

  • 3 years ago

Let α,β be the roots of x^2+112x+1, let γ,δ be the roots of x^2+139x+1. What is the value of (α−γ)(α−δ)(β−γ)(β−δ)?

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  1. Liothique
    • 3 years ago
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    There must be a "trick" I don't see . The roots are too complicated to calculate/simplify manually. For example, delta is \[ \frac{1}{2} \left(-139+\sqrt{19317}\right) \]

  2. mukushla
    • 3 years ago
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    \[(α−γ)(α−δ)(β−γ)(β−δ)=(α^2-(γ+δ)α+γδ) \ (β^2-(γ+δ)β+γδ)\] put the values of \(γ+δ\) and \(γδ\) then simplify

  3. hartnn
    • 3 years ago
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    \(\alpha^2+112\alpha+1=0\) \(\beta^2+112\beta+1=0\) \(\alpha\beta=1,\alpha+\beta=112\) \(\gamma^2+139\gamma+1=0\) \(\delta^2+139\delta+1=0\) \(\gamma+\beta=139,\gamma\beta=1\) \((\alpha-\gamma)(\alpha-\delta)=(\alpha^2-(\gamma+\delta)\alpha+\gamma\delta)\) does this lead u to somewhere ?

  4. mukushla
    • 3 years ago
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    \[γ+δ=-139\]\[γδ=1\]

  5. mukushla
    • 3 years ago
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    \[(α−γ)(α−δ)(β−γ)(β−δ)=(α^2-(γ+δ)α+γδ) \ (β^2-(γ+δ)β+γδ)\]\[=(α^2+139α+1) \ (β^2+139β+1)\]

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