If z -5-3 i and the value of z 4-9 z 3-26 z 2-14 z -8 is k- where k - R- then the value of - k is

Z= 5+3i ⇒ (z+5)=3i ⇒ (z+5)^2= 9 ⇒ z^2+10z+34=0 ⋯ (1) Now using equation (1), we get z^4+9z^3+26z^2 14z+8 nbsp;= z^2(z^2+10z+34) z^3 8z^2 14z+8 nbsp;= 0 ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Complex Numbers

If z =-5+3 i ...

Question

If $z = - 5 + 3 i$ and the value of $z^{4} + 9 z^{3} + 26 z^{2} - 14 z + 8$ is $k$ , where $k \in R$ , then the value of $- k$ is

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Solution

$z = - 5 + 3 i \Rightarrow (z + 5) = 3 i \Rightarrow (z + 5)^{2} = - 9 \Rightarrow z^{2} + 10 z + 34 = 0 \dots (1)$

Now using equation $(1)$ , we get
$z^{4} + 9 z^{3} + 26 z^{2} - 14 z + 8 = z^{2} (z^{2} + 10 z + 34) - z^{3} - 8 z^{2} - 14 z + 8 = 0 - z (z^{2} + 10 z + 34) + 2 z^{2} + 20 z + 8 = 2 z^{2} + 20 z + 8 = 2 (z^{2} + 10 z + 34) - 60 = - 60 \Rightarrow k = - 60$
$∴ - k = 60$

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Algebra of Complex Numbers

Standard XII Mathematics

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If z=−5+3i and the value of z4+9z3+26z2−14z+8 is k, where k∈R, then the value of −k is

z=−5+3i⇒(z+5)=3i⇒(z+5)2=−9⇒z2+10z+34=0⋯(1) Now using equation (1), we get z4+9z3+26z2−14z+8=z2(z2+10z+34)−z3−8z2−14z+8=0−z(z2+10z+34)+2z2+20z+8=2z2+20z+8=2(z2+10z+34)−60=−60⇒k=−60 ∴−k=60

If $z = - 5 + 3 i$ and the value of $z^{4} + 9 z^{3} + 26 z^{2} - 14 z + 8$ is $k$ , where $k \in R$ , then the value of $- k$ is