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

Cheyanne Leigh 2021-09-07 Answered
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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Expert Answer

Maciej Morrow
Answered 2021-09-08 Author has 14979 answers

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

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