Complex Complex

**banku12** · March 17th 2010, 11:50 PM

if a is a complex number which satisfy $ia^3+a^2-a+1=0$

then find $\left | a \right |$ ?

one way is to put a=x+iy

any other short way ?

**Soroban** · March 18th 2010, 08:10 AM

Hello, banku12!

Is there a typo?
If the last term is $i$ , I can solve it.

If a is a complex number which satisfies: . $ia^3+a^2-a + {\color{red}i}\:=\:0$

then find: . $|a|$

Divide by $i\!:\;\;a^3 + \frac{a^2}{i} - \frac{a}{i} + \frac{i}{i} \:=\:0 \quad\Rightarrow\quad a^3 - ia^2 + ia + 1 \:=\:0$

$\begin{array}{ccccc}\text{We have:} & a^3 - ia^2 + ia - i^2 &=& 0 \\ \\<br /> <br /> \text{Factor:} & a^2(a-i) + i(a-i) &=& 0 \\ \\<br /> <br /> \text{Factor:} & (a-i)(a^2+i) &=& 0 \end{array}$

. . Hence: . $a \;=\;i,\;\pm\sqrt{i}$

Therefore, for all roots: . $|a|\:=\:1$

**banku12** · March 18th 2010, 09:17 AM

No..actually there is no typo..

btw is it not possible to compute mod a if last term in the given exp is 1 ?

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