Answer to Question #117227 in Algebra for Vincent Zokah

Question

Given that 3 + i is a root of the equation z
3 3 3z
2 2 8z + 30 = 0, find the
remaining roots.

Answer 1

Given equation is $z^3 - 3z^2 -8z +30=0$ .

Also given $z=3+i$ is root of given equation, so conjugate of given root is also root of the given equation.

Thus $z=3-i$ is root of equation.

So, $z-3-i \ and \ z-3+i$ are two factor of given equation

$\implies (z-3-i)(z-3+i) = (z-3)^2 - i^2 = z^2 - 6z +9 +1 = z^2 -6z+10$ is factor of given equation.

Now dividing $z^3 - 3z^2 -8z +30=0$ by $z^2 -6z+10$ , we got quotient is $z+3$ .

So, another root of given equation is $z=-3.$

Answer: Remaining root of given equation are $z= 3-i, -3.$