if a is a complex number which satisfy $\displaystyle ia^3+a^2-a+1=0$
then find $\displaystyle \left | a \right |$?
one way is to put a=x+iy
any other short way ?
Hello, banku12!
Is there a typo?
If the last term is $\displaystyle i$, I can solve it.
If a is a complex number which satisfies: .$\displaystyle ia^3+a^2-a + {\color{red}i}\:=\:0$
then find: .$\displaystyle |a|$
Divide by $\displaystyle i\!:\;\;a^3 + \frac{a^2}{i} - \frac{a}{i} + \frac{i}{i} \:=\:0 \quad\Rightarrow\quad a^3 - ia^2 + ia + 1 \:=\:0$
$\displaystyle \begin{array}{ccccc}\text{We have:} & a^3 - ia^2 + ia - i^2 &=& 0 \\ \\
\text{Factor:} & a^2(a-i) + i(a-i) &=& 0 \\ \\
\text{Factor:} & (a-i)(a^2+i) &=& 0 \end{array}$
. . Hence: .$\displaystyle a \;=\;i,\;\pm\sqrt{i}$
Therefore, for all roots: .$\displaystyle |a|\:=\:1$