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)?

Question

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)?

anonymous · Answer

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

anonymous · Answer

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

hartnn · Answer

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

anonymous · Answer

$$γ+δ=-139$$$$γδ=1$$

anonymous · Answer

$$(α−γ)(α−δ)(β−γ)(β−δ)=(α^2-(γ+δ)α+γδ) \ (β^2-(γ+δ)β+γδ)$$$$=(α^2+139α+1) \ (β^2+139β+1)$$