1) if
z1,z2,z3,z4
are roots of equation
a0z4+a1z3+a2z2+a3z+a4=0
1)
are also roots of the equation
2)z1 is equal to atleast one of
3)
are also roots of equation
4) none of these multiple correct
please give a detailed explanation on the answers u give [1]
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3 Answers
62the answer depends on ar are real or imaginary...
if they are then the answer is 1, 2
otherwise it is none of these...
62one simple way to think (without proof) is that roots are conjugates of each other..
so one of the zi 's has to be equal to another zi's conjugate... (or itself if it is real!)
this shows 1 and 2
3 is not true because if z is a root, then -z may or may not be a root!
106replace z by z
then the equation becomes aoz4 + a1z3 + a2z2 + a3z + a4 = 0
now ai = ai
So, we get aoz4 + a1z3 + a2z2 + a3z + a4 = 0
which is obiously true if z satisfies the equation
now, a biquadratic equation has max 4 roots and as z1, z2, z3 and z4 are 4 roots (and as zi are also roots), it follows that z1 must be equal to atleast one of zi
further as coefficients are real, complex roots occur in pairs. So z1 and z4 (say) must be conjugates of each other. hence z1 = z4
