What is the HCF polynomial, ${{x}^{4~}}-~3x~+2,~{{x}^{3}}-~3{{x}^{2~}}+~3x~-1$ and ${{x}^{4~}}-1?$
Solution
Let $p\left( x \right)=~{{x}^{4~}}-~3x~+~2 = ~\left( x~-~1 \right)\left( {{x}^{3~}}+~{{x}^{2~}}+~x-2 \right)$
$q\left( x \right)=~{{x}^{3}}-~3{{x}^{2~}}+~3x~-1 =~{{\left( x~-~1 \right)}^{3}}$
and, $r\left( x \right)=~{{x}^{4~}}-~1~=~\left( {{x}^{2~}}-~1 \right)\left( {{x}^{2~}}+~1 \right)=~\left( x~-1 \right)\left( x~+1 \right)\left( {{x}^{2~}}+~1 \right)$
Therefore, HCF will be $=~x~-~1$