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For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $

$1)$ show that $2+i $ is a root.

$2)$ solve the given equation.

Attemp to solve: I'm not really sure how to solve this, but I considered using Vietes formulas to comstruct a system of equations with two unknowns, and from there to cacluate the roots.....I'm not sure how to do No. 1 though...

help with this would be appricated...

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Here's the steps:

  • check that $2+i$ is a root.
  • identify the coefficients $b$ and $c$ such that the given polynomial be equal to $$(z-2-i)(z^2+bz+c)$$
  • find the roots of the quadratic polynomial using the discriminant $\Delta$.
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  • $\begingroup$ how does one perform this check? $\endgroup$ – Bak1139 Jul 15 '14 at 14:17
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    $\begingroup$ Replace $z$ by $2+i$ in the given polynomial and do the calculus to get $0$. $\endgroup$ – user63181 Jul 15 '14 at 14:18
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Just use long division

enter image description here

Now we have the problem simplified to one of $$(z-(2+i))(z^2+(-3+i)z-3i)=0$$ Now it's just a matter of applying the Quadratic Formula to find the other 2 roots

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