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