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Show that for any real numbers `a_(3),a_(4),a_(5),……….a_(85)`, the roots of the equation
`a_(85)x^(85)+a_(84)x^(84)+……….+a_(3)x^(3)+3x^(2)+2x+1=0` are not real.

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Let `P(x)=a_(85)x^(85)+a_(84)x^(84)`
`+…..+a_(3)x^(3)+3x^(2)+2x+1=0`…..i
Since `P(0)=1` then 0 is not a root fo eq. I
Let `alpha_(1), alpha_(2), alpha_(3), …, alpha_(85)` be the complex roots of Eq. (i)
Then the `beta_(i)("let"1/(alpha_(i)))` the complex roots of the polynomial
`Q(y)=y^(85)+2y^(84)+3y^(83)+a_(3)y^(82)+..........+a_(85)`
It follows that
`sum_(i=1)^(85)beta_(i)=-2` and `sum_(1leiltjle85)sumbeta_(i)beta_(j)=3`
Then `sum_(i=1)^(85)beta_(i)^(2)=(sum_(i=1)^(85)beta_(i))^(2)-2sum_(1leiltjle85)sumbeta_(i)beta_(j)`
`=4-6=-2lt0`
Then the `beta_(i)'s` is not all real and then `alpha_(i)`'s are not all real.
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