n:(z-1+i)^4-5(z-1+i)^2+4=0 କୁ ସମାଧାନ କରନ୍ତୁ

$$n : ( z - 1 + i ) ^ { 4 } - 5 ( z - 1 + i ) ^ { 2 } + 4 = 0$$

n ପାଇଁ ସମାଧାନ କରନ୍ତୁ

$$n=\left(5\left(z+\left(-1+i\right)\right)^{2}-4\right)\left(z+\left(-1+i\right)\right)^{4}$$

$$z\neq 1-i$$

ରୈଖିକ ସମୀକରଣ ସମାଧାନ କରିବା ପାଇଁ ପଦକ୍ଷେପଗୁଡିକ

ସମୀକରଣ ଉଭୟ ପାର୍ଶ୍ୱକୁ z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ଗୁଣନ କରନ୍ତୁ.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5\left(z-1+i\right)^{2}\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

ବର୍ଗ z-1+i.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5\left(z^{2}+\left(-2+2i\right)z-2i\right)\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

-5 କୁ z^{2}+\left(-2+2i\right)z-2i ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5z^{2}+\left(10-10i\right)z+10i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 କୁ \frac{n}{\left(z-1+i\right)^{4}}-5z^{2}+\left(10-10i\right)z+10i ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ ଏବଂ ଏକାପରି ପଦଗୁଡିକୁ ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+150iz^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 କୁ 4 ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+150iz^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+4z^{4}+\left(-16+16i\right)z^{3}-48iz^{2}+\left(32+32i\right)z-16=0$$

\left(4+150i\right)z^{4} ପାଇବାକୁ 150iz^{4} ଏବଂ 4z^{4} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(-16+16i\right)z^{3}-48iz^{2}+\left(32+32i\right)z-16=0$$

\left(-216-184i\right)z^{3} ପାଇବାକୁ \left(-200-200i\right)z^{3} ଏବଂ \left(-16+16i\right)z^{3} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-48iz^{2}+\left(32+32i\right)z-16=0$$

\left(300-48i\right)z^{2} ପାଇବାକୁ 300z^{2} ଏବଂ -48iz^{2} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(32+32i\right)z-16=0$$

\left(-88+152i\right)z ପାଇବାକୁ \left(-120+120i\right)z ଏବଂ \left(32+32i\right)z ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=0$$

ଉଭୟ ପାର୍ଶ୍ଵକୁ 5z^{6} ଯୋଡନ୍ତୁ. ଯାହାକିଛି ସହିତ ଶୂନ୍ୟ ଯୋଗ ହେଲେ ସେହି ସଂଖ୍ୟା ମିଳିଥାଏ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}$$

ଉଭୟ ପାର୍ଶ୍ୱରୁ \left(30-30i\right)z^{5} ବିୟୋଗ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}$$

ଉଭୟ ପାର୍ଶ୍ୱରୁ \left(4+150i\right)z^{4} ବିୟୋଗ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}$$

ଉଭୟ ପାର୍ଶ୍ୱରୁ \left(-216-184i\right)z^{3} ବିୟୋଗ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}$$

ଉଭୟ ପାର୍ଶ୍ୱରୁ \left(300-48i\right)z^{2} ବିୟୋଗ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}$$

ଉଭୟ ପାର୍ଶ୍ୱରୁ \left(-88+152i\right)z ବିୟୋଗ କରନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z$$

ଉଭୟ ପାର୍ଶ୍ଵକୁ 40i ଯୋଡନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z+40i$$

ଉଭୟ ପାର୍ଶ୍ଵକୁ 16 ଯୋଡନ୍ତୁ.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z+40i+16$$

ସମୀକରଣ ଉଭୟ ପାର୍ଶ୍ୱକୁ z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

-30+30i ପ୍ରାପ୍ତ କରିବାକୁ -1 ଏବଂ 30-30i ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

-4-150i ପ୍ରାପ୍ତ କରିବାକୁ -1 ଏବଂ 4+150i ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

216+184i ପ୍ରାପ୍ତ କରିବାକୁ -1 ଏବଂ -216-184i ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

-300+48i ପ୍ରାପ୍ତ କରିବାକୁ -1 ଏବଂ 300-48i ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

88-152i ପ୍ରାପ୍ତ କରିବାକୁ -1 ଏବଂ -88+152i ଗୁଣନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}+\left(88-152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 କୁ 5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}+\left(88-152i\right)z ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ ଏବଂ ଏକାପରି ପଦଗୁଡିକୁ ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8360i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 କୁ 40i ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8360i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+40iz^{4}+\left(-160-160i\right)z^{3}+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

\left(1104+8400i\right)z^{4} ପାଇବାକୁ \left(1104+8360i\right)z^{4} ଏବଂ 40iz^{4} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+\left(-160-160i\right)z^{3}+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

\left(-5632-3968i\right)z^{3} ପାଇବାକୁ \left(-5472-3808i\right)z^{3} ଏବଂ \left(-160-160i\right)z^{3} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

\left(3600-704i\right)z^{2} ପାଇବାକୁ \left(3120-704i\right)z^{2} ଏବଂ 480z^{2} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-352+608i\right)z+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

\left(-672+928i\right)z ପାଇବାକୁ \left(-352+608i\right)z ଏବଂ \left(-320+320i\right)z ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 କୁ 16 ଦ୍ୱାରା ଗୁଣନ କରିବା ପାଇଁ ବିତରଣାତ୍ମକ ଗୁଣଧର୍ମ ବ୍ୟବହାର କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+16z^{4}+\left(-64+64i\right)z^{3}-192iz^{2}+\left(128+128i\right)z-64$$

\left(1120+8400i\right)z^{4} ପାଇବାକୁ \left(1104+8400i\right)z^{4} ଏବଂ 16z^{4} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+\left(-64+64i\right)z^{3}-192iz^{2}+\left(128+128i\right)z-64$$

\left(-5696-3904i\right)z^{3} ପାଇବାକୁ \left(-5632-3968i\right)z^{3} ଏବଂ \left(-64+64i\right)z^{3} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i-192iz^{2}+\left(128+128i\right)z-64$$

\left(3600-896i\right)z^{2} ପାଇବାକୁ \left(3600-704i\right)z^{2} ଏବଂ -192iz^{2} ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-672+928i\right)z-160i+\left(128+128i\right)z-64$$

\left(-544+1056i\right)z ପାଇବାକୁ \left(-672+928i\right)z ଏବଂ \left(128+128i\right)z ସମ୍ମେଳନ କରନ୍ତୁ.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z-160i-64$$

n ଧାରଣ କରିଥିବା ସମସ୍ତ ପଦ ସମ୍ମେଳନ କରନ୍ତୁ.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z-160i-64$$

ସମୀକରଣ ମାନାଙ୍କ ରୂପରେ ରହିଛି.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)$$

ଉଭୟ ପାର୍ଶ୍ୱକୁ z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ବିଭାଜନ କରନ୍ତୁ.

$$\frac{\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}=\frac{5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}$$

z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ବିଭାଜନ କରିବା z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ଗୁଣନକୁ ପୂର୍ବବତ୍‌ କରିଥାଏ.

$$n=\frac{5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}$$

5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-544+1056i\right)z+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-64-160i\right) କୁ z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 ଦ୍ୱାରା ବିଭାଜନ କରନ୍ତୁ.

$$n=\left(5z^{2}+\left(-10+10i\right)z+\left(-4-10i\right)\right)\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)$$