6. TBQ - Find the pattern in the expansion

Expand (i) \((u – v) (u + v)\), (ii) \((u – v) (u^{2}+ uv + v^{2})\) and (iii) \((u – v)(u^{3}+ u^{2}v + uv^{2}+ v^{3})\), Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

 

 

 (i) \((u – v) (u + v)\) \(=\) $i^i-i^i$

 

(ii) \((u – v) (u^{2}+ uv + v^{2})\) \(=\) $i^i-i^i$

 

(iii) \((u – v)(u^{3}+ u^{2}v + uv^{2}+ v^{3})\) \(=\) $i^i-i^i$

 

Do you see a pattern in the above expansion? YesNo

 

The next identity in the pattern is $\left(u-v\right)\left(i^i+i^{i}i+i^i{i}^i+i{i}^i-i^i\right)$.

 

The expansion of the above found identity is $i^i-i^i$.