# Factor each of the following polynomials as the product of two polynomials of degree 1 in Z_{10}[x]. x+9

Factor each of the following polynomials as the product of two polynomials of degree 1 in $$\displaystyle{Z}_{{{10}}}{\left[{x}\right]}$$.
$$\displaystyle{x}+{9}$$

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Maciej Morrow

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 $$\displaystyle{Z}_{{{10}}}{\left[{x}\right]}$$.