Solving the complex polynomial

Question

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...

Answer 1

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 $\Delta$.

Answer 2

Just use long division

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