To factor the polynomial \(\displaystyle{x}+{9}\) as the product of two polynomials of degree in \(\displaystyle{Z}_{{{10}}}{\left[{x}\right]}\)

Consider two polynomials \(\displaystyle{\left({5}{x}+{3}\right)}\) and \(\displaystyle{\left({2}{x}+{3}\right)}\) of degree 1 in \(\displaystyle{Z}_{{{10}}}{\left[{x}\right]}\)

The product of the two polynomials is given as,

\(\displaystyle{\left({5}{x}+{3}\right)}{\left({2}{x}+{3}\right)}={\left({5}\times{2}\right)}{x}^{{2}}+{\left({5}\times{3}\right)}{x}+{\left({3}\times{2}\right)}{x}+{9}\)

\(\displaystyle={10}{x}^{{2}}+{15}{x}+{6}{x}+{9}\)

\(\displaystyle={0}.{x}^{{2}}+{\left({15}+{6}\right)}{x}+{9}\)

\(\displaystyle={0}.{x}^{{2}}+{21}{x}+{9}\)

\(\displaystyle={x}+{9}\)

since \(\displaystyle{10}={0},{21}={1}\) in \(\displaystyle{Z}_{{{10}}}{\left[{x}\right]}\)

Therefore, \(x+9\) is dactored as a product of two polynomials \((5x+3)\) and \((2x+3)\) of degree 1 in \(\displaystyle{Z}_{{{10}}}{\left[{x}\right]}\).