Question

if z =-3+√-2 , then prove that z⁴ + 5z³ + 8z² +7z +4 is equal to -29 ​

Answer 1

Chapter- Complex Numbers

Given: $\mathsf{z=-3+\sqrt{2}i}$

To find: $\mathsf{z^{4}+5z^{3}+8z^{2}+7z+4}$

Solution:

• Remember: $\sqrt{-1}=i$

• Given, $\mathsf{z=-3+\sqrt{2}i}$

• Then, $\mathsf{z^{2}=(-3+\sqrt{2}i)^{2}}$
• $\mathsf{=9-6\sqrt{2}i+2i^{2}}$
• $\mathsf{=9-6\sqrt{2}i-2}$
• $\mathsf{=7-6\sqrt{2}i}$

• Also, $\mathsf{z^{3}=z\times z^{2}}$
• $\mathsf{=(-3+\sqrt{2}i)(7-6\sqrt{2}i)}$
• $\mathsf{=-21+18\sqrt{2}i+7\sqrt{2}i-12i^{2}}$
• $\mathsf{=-21+25\sqrt{2}i+12}$
• $\mathsf{=-9+25\sqrt{2}i}$

• And $\mathsf{z^{4}=(z^{2})^{2}}$
• $\mathsf{=(7-6\sqrt{2}i)^{2}}$
• $\mathsf{=49-84\sqrt{2}i+72i^{2}}$
• $\mathsf{=49-84\sqrt{2}i-72}$
• $\mathsf{=-23-84\sqrt{2}i}$

Therefore, $\mathsf{z^{4}+5z^{3}+8z^{2}+7z+4}$

$\mathsf{=-23-84\sqrt{2}i-45+125\sqrt{2}i}$

$\quad\mathsf{+56-48\sqrt{2}i-21+7\sqrt{2}i+4}$

$\mathsf{=-29}$

Thus proved.