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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

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

$(α−γ)(α−δ)(β−γ)(β−δ)=(α^2-(γ+δ)α+γδ) \ (β^2-(γ+δ)β+γδ)$ put the values of $$γ+δ$$ and $$γδ$$ then simplify

3. hartnn

$$\alpha^2+112\alpha+1=0$$ $$\beta^2+112\beta+1=0$$ $$\alpha\beta=1,\alpha+\beta=112$$ $$\gamma^2+139\gamma+1=0$$ $$\delta^2+139\delta+1=0$$ $$\gamma+\beta=139,\gamma\beta=1$$ $$(\alpha-\gamma)(\alpha-\delta)=(\alpha^2-(\gamma+\delta)\alpha+\gamma\delta)$$ does this lead u to somewhere ?

4. mukushla

$γ+δ=-139$$γδ=1$

5. mukushla

$(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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