# how to prove z0 is a root of the polynomial

• Nov 1st 2012, 09:40 AM
flametag3
how to prove z0 is a root of the polynomial
How can I prove that $z_0$ and $\overline{z_{0}}$ is a root of the polynomial
$a_{n}z^n + a_{(n-1)} z^{(n-1)} +....+a_{1} z + a_{0} = 0$

Using $(\bar{z})^k = \overline{z^k}$

I have been able to prove $(\bar{z})^k = \overline{z^k}$ but I dont get how to use that to show that $z_0$ and $\overline{z_{0}}$ are roots of the polynomial.

• Nov 1st 2012, 06:16 PM
chiro
Re: how to prove z0 is a root of the polynomial
Hey flametag3.

Is this asking to show that if z0 is a root then z0_bar is a root or is it asking that if the condition (z_bar)^k = (z^k)_bar, then that particular z is a root for when that is true?
• Nov 2nd 2012, 09:08 PM
flametag3
Re: how to prove z0 is a root of the polynomial
Hey Chiro,

thanks for your reply. Its asking to show that if z0 is a root then z0_bar is a root as well.
• Nov 2nd 2012, 10:35 PM
chiro
Re: how to prove z0 is a root of the polynomial
You'll have to show that if z gives 0 then z_bar also gives 0, and one to show that given what you have is to show that the equation with z_bar is zero if the one with z is zero and you will need to apply this term by term in your polynomial to show that both equal zero and thus both are roots of the polynomial if they have a specific f(z) form (note f(z) doesn't change but z = z or z = z_bar both need to give f(z) = 0 to both be roots).