Factorize exercise asked by Buddhadev Konai in the math group

Exercise:

Factorize ${{x}^{5}}+x+1$

Solution: ${{x}^{5}}+x+1={{x}^{5}}-{{x}^{2}}+{{x}^{2}}+x+1={{x}^{2}}\left( {{x}^{3}}-1 \right)+{{x}^{2}}+x+1$

But ${{x}^{3}}-1=\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)$ so ${{x}^{5}}+x+1={{x}^{2}}\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)+{{x}^{2}}+x+1$

 $=\left( {{x}^{2}}+x+1 \right)\left( {{x}^{2}}\left( x-1 \right)+1 \right)=\left( {{x}^{2}}+x+1 \right)\left( {{x}^{3}}-{{x}^{2}}+1 \right)$