If z 1- z 2- z 3- z 4 are 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- thenA- z1- z2- z3- z4 are also roots of the equationB- z 1 is equal to at least two of z1- z2- z3- z4C- -z1-z2-z3-z4 are also roots of the equationD- none of these

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Properties of Conjugate of a Complex Number

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}, z_{4}$ are roots of the equation $a_{0} z^{4} + a_{1} z^{3} + a_{2} z^{2} + a_{3} z + a_{4} = 0$ where $a_{0}, a_{1}, a_{2}, a_{3} a n d a_{4}$ are real, then

are also roots of the equation

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z₁ is equal to at least two of

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are also roots of the equation

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none of these

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Solution

The correct option is A are also roots of the equation
$a_{0} z^{4} + a_{1} z^{3} + a_{2} z^{2} + a_{3} z + a_{4} = 0$

$\Rightarrow a_{0} {¯ z}^{4} + a_{1} {¯ z}^{3} + a_{2} {¯ z}^{2} + a_{3} ¯ z + a_{4} = 0$ (taking congugate on both sides)

$\Rightarrow a_{0} (¯ z)^{4} + a_{1} (¯ z)^{3} + a_{2} (¯ z)^{2} + a_{3} ¯ z + a_{4} = 0$

$\Rightarrow ¯ z$ is a root of the equation if z is a root. so, option(A) is correct.

Also if $z_{1}$ is real, $z_{1} =_{1} .$
If $z_{1}$ is non real complex then $_{1}$ is also a root because imaginary roots occur in conjugate pairs.

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