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Question

if$$\displaystyle\ z_{1},z_{2},z_{3},z_{4}$$ are the roots of the equation $$\displaystyle\ a_{0}z^{4}+a_{1}z^{3}+a_{2}z^{2}+a_{3}z+a_{4}=0$$ where $$\displaystyle\ a_{0},a_{1},a_{2},a_{3}$$ and$$\displaystyle\ a_{4}$$ are real, then


A
 ¯z1, ¯z2,, ¯z3,  ¯z4 are also the roots.
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B
 z1 is equal to at least one of  ¯z1, ¯z2, ¯z3, ¯z4
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C
 ¯z1,¯z2,¯z3,¯z4 are also roots of the equation
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D
none of these
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Solution

The correct options are
A $$\displaystyle\ \bar{z_{1}}$$,$$\displaystyle\ \bar{z_{2}},$$,$$\displaystyle\ \bar{z_{3}}$$, $$\displaystyle\ \bar{z_{4}}$$ are also the roots.
B $$\displaystyle\ z_{1}$$ is equal to at least one of $$\displaystyle\ \bar{z_{1}}$$,$$\displaystyle\ \bar{z_{2}}$$,$$\displaystyle\ \bar{z_{3}}$$,$$\displaystyle\ \bar{z_{4}}$$
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
$$z_{a},\bar{z_{a}}$$ and $$z_{b},\bar{z_{b}}$$.
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.

Mathematics

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