I had a question in my homework that I managed to solve part A of it with one of the Vieta's formulas, but part B was wrong. I am almost sure that part B also uses one of the Vieta's formulas but I just don't know how to use it properly. The question is:

P(z)=z^7 + a6*z^6 + a5*z^5 + a4*z^4 + a3*z^3 + a2*z^2 +a1*z + a0

provided information:

z1=1+2i is the root of P(z),P'(z) and P''(z) ;

a0=-53

A) Real root of P(z) is _____ ?

B) a6=?

My answer to A was :

from the information we can say that (1+2i) is root of multiplicity 3. and because it's a complex number also the conjugate is of multiplicity 3.

and the only one left is a real (because of 7 roots) so I mark it as 't' here.

So I used Vieta's formula for multiplying and got:

(1+2i)^3*(1-2i)^3*t = (-1)^7*(a0/a7) => // a0=-53 ; a7=1

=> 125*t = (-1)*(-53/1)

=>t = 53/125

This is part A and is correct.

Now what I tried to do for part B was to use the Vieta's sum formula in this way:

The sum of complex roots is 0, and the remaining root is 53/125.

So:

0+53/125 = -a6/a7 => // a7=1

a6=-53/125 < But that's a wrong answer.

What am I doing wrong?

Thanks!