if z is acomplex number that satisfies the equation z^4+z^3+2z^2+z+1=0,then find the value of the conjugate of z

Question

perl · Answer

so assuming z is a solution, we want z* the conjugate

perl · Answer

if  f (z ) = 0,  then f ( z* ) = 0

perl · Answer

we might be able to multiply 
( z^4+z^3+2z^2+z+1) * ( z'^4+z'^3+2z'^2+z'+1) = 0 * 0

perl · Answer

i will put z = a + bi, and z ' = a - bi , into wolfram, and see what it says

perl · Answer

z= i, and z  = -i  solve the equation

anonymous · Answer

yeah when i substitute z as i ,the equation is right.....but how do you figure it up?

perl · Answer

perl · Answer

but i think we should use properties of complex numbers, do you have any notes on them

anonymous · Answer

i don't

zarkon · Answer

$$z^4+z^3+2z^2+z+1=z^4+z^3+z^2+z^2+z+1$$

$$(z^4+z^3+z^2)+(z^2+z+1)=z^2(z^2+z+1)+(z^2+z+1)$$

$$=(z^2+1) (z^2+z+1)$$