0
$\begingroup$

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...

$\endgroup$
2
$\begingroup$

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 $\Delta$.
$\endgroup$
  • $\begingroup$ how does one perform this check? $\endgroup$ – Bak1139 Jul 15 '14 at 14:17
  • 1
    $\begingroup$ Replace $z$ by $2+i$ in the given polynomial and do the calculus to get $0$. $\endgroup$ – user63181 Jul 15 '14 at 14:18
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.