Question

if$z_1,z_2,z_3,z_4$ are the roots of the equation $a_0{z}^4+a_1{z}^3+a_2{z}^2+a_{3}z+a_4=0$ where $a_0,a_1,a_2,a_3$ and$a_4$ are real, then

• A. $\overline{z_1}$,$\overline{z_2},$,$\overline{z_3}$, $\overline{z_4}$ are also the roots.
• B. $z_1$ is equal to at least one of $\overline{z_1}$,$\overline{z_2}$,$\overline{z_3}$,$\overline{z_4}$
• C. $-\overline{z_1},-\overline{z_2},-\overline{z_3},\overline{z_4}$ are also roots of the equation
• D. none of these

Answer 1

Answer: (A), (B)

The correct options are
A $\overline{z_1}$,$\overline{z_2},$,$\overline{z_3}$, $\overline{z_4}$ are also the roots.
B $z_1$ is equal to at least one of $\overline{z_1}$,$\overline{z_2}$,$\overline{z_3}$,$\overline{z_4}$

If in an equation the coefficients are real, then if roots of the equation are complex, they will occur as conjugate pairs.
Hence
$z_1$ is conjugate of atleast one of the roots.
Similarly $z_2$ is conjugate of atleast one of the roots.
In total we get 2 pair of conjugate roots
$z_a,\overline{z_a}$ and $z_b,\overline{z_b}$.
Hence if $z_1,z_2,z_3,z_4$ are the roots of the given equation, then their conjugates are also the roots of the given equation.
Hence both option A and B are true.