Blog: `(x^2 + x + 2)/(x^2 + 2)^2` Write the partial fraction decomposition of the rational expression.

Wednesday, July 8, 2015

$\displaystyle\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$ Write the partial fraction decomposition of the rational expression.

$\displaystyle\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$

Let $\displaystyle\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}=\frac{{{A}{x}+{B}}}{{{{x}}^{{2}}+{2}}}+\frac{{{C}{x}+{D}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$

$\displaystyle\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}=\frac{{{\left({A}{x}+{B}\right)}{\left({{x}}^{{2}}+{2}\right)}+{C}{x}+{D}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$

$\displaystyle\therefore{\left({{x}}^{{2}}+{x}+{2}\right)}={\left({A}{x}+{B}\right)}{\left({{x}}^{{2}}+{2}\right)}+{C}{x}+{D}$

$\displaystyle{{x}}^{{2}}+{x}+{2}={A}{{x}}^{{3}}+{2}{A}{x}+{B}{{x}}^{{2}}+{2}{B}+{C}{x}+{D}$

$\displaystyle{{x}}^{{2}}+{x}+{2}={A}{{x}}^{{3}}+{B}{{x}}^{{2}}+{\left({2}{A}+{C}\right)}{x}+{2}{B}+{D}$

Equating coefficients of like terms gives,

$\displaystyle{A}={0}$

$\displaystyle{B}={1}$

$\displaystyle{2}{A}+{C}={1}$

$\displaystyle{2}{B}+{D}={2}$

substituting the values of A and B in the above equations,

$\displaystyle{2}{\left({0}\right)}+{C}={1}$

$\displaystyle{C}={1}$

$\displaystyle{2}{\left({1}\right)}+{D}={2}$

$\displaystyle{2}+{D}={2}$

$\displaystyle{D}={2}-{2}$

$\displaystyle{D}={0}$

$\displaystyle\therefore\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}=\frac{{1}}{{{{x}}^{{2}}+{2}}}+\frac{{x}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$

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Wednesday, July 8, 2015

$\displaystyle\frac{{{{x}}^{{2}}+{x}+{2}}}{{{\left({{x}}^{{2}}+{2}\right)}}^{{2}}}$ Write the partial fraction decomposition of the rational expression.

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