# Math Help - Complex numbers

1. ## Complex numbers

2. Originally Posted by norivea
1a) Find the real and complex roots of:

$
(z-3)(z^2-5z+8)=0
$

This has three roots, one real corresponding to the first factor; so $x=3$ is a root.

The other two roots are the roots of the second factor $z^2-5z+8=0$, these may
be found using the quadratic formula, which gives in this case:

$
z=\frac{5}{2}\pm \frac{\sqrt{7}}{2}i
$

RonL

3. Originally Posted by norivea
1b) Find the real and complex roots of:

$
z^3-10z^2+34z-40
$

given that $3-i$ is a root.

A cubic with real coefficients always has at least one real root, and
complex roots occur in conjugate pairs. So both $3-i$ and
$3+i$ are roots. Also:

$
z^3-10z^2+34z-40=(z-(3-i))(z-(3+i))(z-a)
$

where $a$ is real, and so easily found to be $4$,
so the remaining root is $z=4$.

RonL