Answer to Question #117180 in Algebra for Edward

Question

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

Answer 1

The equation is given as:

$z^3-3z^2-8z+30=0$

Let's type the equation in the form of:

$z^3-6z^2+3z^2+10z-18z+30=0$

After some substitutions:

$z^3-6z^2+10z+3z^2-18z+30=0$

Realease $z$ from the first three components, and 3 from the remaining ones:

$z(z^2-6z+10)+3(z^2-6z+10)=0$

Then, release $z^2-6z+10$ from both multiplications, yields

$(z+3)(z^2-6z+10)=0$

The root of the first bracket is:

$z=-3$ ;

As for the second bracket, we know that complex roots occur in conjugate pairs:

as $3+i$ is a root, so must the other be $3-i$

Now, sum of the roots $=3+i+3-i=6$ and,

the product of the roots $=(3+i)(3-i)=9-3i+3i-i^2=9+1=10$

Hence, the roots of the equation are: $-3; 3-i; 3+i$