Simplifying the sum of the roots

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If the only real root of a fifth degree polynomial is $\frac{1}{3}$ , the sum of all roots in simplified form is:

assumptions

$z_{1}$ and $z_{2}$ are complex roots;
$Re_{(z_{1})}$ means the real part of the complex root $z_{1}$ ;
$Im_{(z_{1})}$ means the imaginary part of the complex root $z_{1}$ .

z_{1}+z_{2}

\mid z_{1} \mid + \mid z_{2} \mid

\frac{1}{3} + 2(Re_{(z_{1})} + Re_{(z_{2})})

\frac{1}{3} + 2(Im_{(z_{1})} + Im_{(z_{2})})

1 solution

Aditya Narayan Sharma
Feb 9, 2016

Since it has only one real root, other four roots are of the form (a+ib),(a-ib),(c+id),(c-id) as complex roots occur in pairs. let z1 = a+ib & z2= c+id be the other twoo roots Therefore Sum = 1/3 + 2(a+c) = 1/3 + 2( Re(z1) + Re(z2) )

Simplifying the sum of the roots

1 solution

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