\(\displaystyle{x}^{{3}}-{6}{x}^{{2}}+{11}{x}-{y}={6}\)

\(\displaystyle{y}={x}^{{3}}-{6}{x}^{{2}}+{11}{x}-{6}\)

\(\displaystyle{p}{\left({x}\right)}={x}^{{3}}-{6}{x}^{{2}}+{11}{x}-{6}\)

\(\displaystyle{p}{\left({1}\right)}={1}-{6}+{11}-{6}={0}\)

(x-1) is a point of p(x)

Dividing p(x) by (x-2) we get

\(\displaystyle{q}{\left({x}\right)}={x}^{{2}}-{3}{x}-{21}{x}+{6}\)

\(\displaystyle{q}{\left({x}\right)}={x}{\left({x}-{3}\right)}-{2}{\left({x}-{3}\right)}\)

\(\displaystyle{q}{\left({x}\right)}={\left({x}-{2}\right)}{\left({x}-{3}\right)}\)

\(\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{q}{\left({x}\right)}\)

\(\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{\left({x}-{2}\right)}{\left({x}-{3}\right)}\)

Point of polynomial are x=1,2,3

\(\displaystyle{y}={x}^{{3}}-{6}{x}^{{2}}+{11}{x}-{6}\)

\(\displaystyle{p}{\left({x}\right)}={x}^{{3}}-{6}{x}^{{2}}+{11}{x}-{6}\)

\(\displaystyle{p}{\left({1}\right)}={1}-{6}+{11}-{6}={0}\)

(x-1) is a point of p(x)

Dividing p(x) by (x-2) we get

\(\displaystyle{q}{\left({x}\right)}={x}^{{2}}-{3}{x}-{21}{x}+{6}\)

\(\displaystyle{q}{\left({x}\right)}={x}{\left({x}-{3}\right)}-{2}{\left({x}-{3}\right)}\)

\(\displaystyle{q}{\left({x}\right)}={\left({x}-{2}\right)}{\left({x}-{3}\right)}\)

\(\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{q}{\left({x}\right)}\)

\(\displaystyle{p}{\left({x}\right)}={\left({x}-{1}\right)}{\left({x}-{2}\right)}{\left({x}-{3}\right)}\)

Point of polynomial are x=1,2,3