# Thread: Complex numbers

1. ## Complex numbers

Please help me answer both of these questions. Thank you 2. Originally Posted by norivea
Please help me answer both of these questions. Thank you 1a) Find the real and complex roots of:

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

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

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

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

RonL

3. Originally Posted by norivea
Please help me answer both of these questions. Thank you 1b) Find the real and complex roots of:

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

given that $\displaystyle 3-i$ is a root.

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

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

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

RonL

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