Question

If z=2, find the value of z⁵-18(z⁴-3z²+z³-10z+30)
please help me out with this problem ​

Answer 1

Answer:

How can I solve this question: If z is a complex number satisfying z⁴ + z³ + 2z² + z + 1 then find |z'|.?

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Let's try to factorise.

>z4+z3+2z2+z+1=0

>z4+z2+z3+z+z2+1=0

>z2(z2+1)+z(z2+1)+1(z2+1)=0

>(z2+1)(z2+z+1)=0

>z2+1=0 or z2+z+1=0

z=±−1−−−√ or z=−1±1−4√2

z=±i or z=−1±i3√2

Therefore, z has the following values

i,−i,−1+i3√2,−1−i3√2

The value of z′ would be

−i,i,−1−i3√2,−1+i3√2

The value of |z′| would be only +1.

Answer 2

$\huge\tt\colorbox{violet}{Solution}$

${z}^{5} - 18( {z}^{4} - 3 {z}^{2} + {z}^{3} - 10z + 30) \\ {2}^{5} - 18( {2}^{4} - 3( {2})^{2} + {2}^{3} - 10(2) \\ + 30 \\ \\ 32 - 18(16 - 12 + 8 - 20 + 30) \\ 14(4 + 8 + 10) \\ 14(22) \\ = 308$