这是一份leeds利兹大学MATH100501作业代写的成功案例
Find all quadratic equations with real coefficients having $1-2 i$ as a root.
As $1-2 i$ is a root, $1+2 i$ is also a root.
Sum of roots $=1-2 i+1+2 i \quad$ Product of roots $=(1-2 i)(1+2 i)$
$\begin{array}{ll}=2 & =1+4 \ & =5\end{array}$
So, as $x^{2}-($ sum $) x+($ product $)=0, \quad=5$
$a\left(x^{2}-2 x+5\right)=0, a \neq 0$ gives all possible equations.
Find exact values of $a$ and $b$ if $\sqrt{2}+i$ is a root of $x^{2}+a x+b=0, a, b \in \mathbb{R}$.
Since $a$ and $b$ are real, the quadratic has real coefficients
$\therefore \sqrt{2}-i$ is also a root
$\therefore$ sum of roots $=\sqrt{2}+i+\sqrt{2}-i=2 \sqrt{2}$
product of roots $=(\sqrt{2}+i)(\sqrt{2}-i)=2+1=3$
Thus $a=-2 \sqrt{2}$ and $b=3$.
MATH100501 COURSE NOTES ：
$\quad$ Find all cubic polynomials with zeros $\frac{1}{2}, \quad-3 \pm 2 i$.
The zeros $-3 \pm 2 i$ have sum $=-3+2 i-3-2 i=-6$ and
$$
\text { product }=(-3+2 i)(-3-2 i)=13
$$
and $\therefore$ come from the quadratic factor $z^{2}+6 z+13$
$\frac{1}{2}$ comes from the linear factor $2 z-1$
$$
\therefore \quad P(z)=a(2 z-1)\left(z^{2}+6 z+13\right), \quad a \neq 0 .
$$