Blog: `(x^3 - x + 3)/(x^2 + x - 2)` Write the partial fraction decomposition of the rational expression.

Friday, July 10, 2015

$\displaystyle\frac{{{{x}}^{{3}}-{x}+{3}}}{{{{x}}^{{2}}+{x}-{2}}}$ Write the partial fraction decomposition of the rational expression.

$\displaystyle\frac{{{{x}}^{{3}}-{x}+{3}}}{{{{x}}^{{2}}+{x}-{2}}}$

Divide by applying long division method,

$\displaystyle\frac{{{{x}}^{{3}}-{x}+{3}}}{{{{x}}^{{2}}+{x}-{2}}}={\left({x}-{1}\right)}+\frac{{{2}{x}+{1}}}{{{{x}}^{{2}}+{x}-{2}}}$

Now continue with the partial fraction of the remainder expression,

$\displaystyle\frac{{{2}{x}+{1}}}{{{{x}}^{{2}}+{x}-{2}}}=\frac{{{2}{x}+{1}}}{{{{x}}^{{2}}-{x}+{2}{x}-{2}}}$

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

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

Let $\displaystyle\frac{{{2}{x}+{1}}}{{{{x}}^{{2}}+{x}-{2}}}=\frac{{A}}{{{x}-{1}}}+\frac{{B}}{{{x}+{2}}}$

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

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

$\displaystyle\therefore{\left({2}{x}+{1}\right)}={A}{x}+{2}{A}+{B}{x}-{B}$

$\displaystyle{2}{x}+{1}={x}{\left({A}+{B}\right)}+{2}{A}-{B}$

Equating the coefficients of the like terms,

$\displaystyle{A}+{B}={2}$ ---- equation 1

$\displaystyle{2}{A}-{B}={1}$ --- equation 2

Add the equation 1 and 2,

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

$\displaystyle{3}{A}={3}$

$\displaystyle{A}={1}$

Plug the value of A in the equation 1,

$\displaystyle{1}+{B}={2}$

$\displaystyle{B}={2}-{1}$

$\displaystyle{B}={1}$

$\displaystyle\therefore\frac{{{{x}}^{{3}}-{x}+{3}}}{{{{x}}^{{2}}+{x}-{2}}}={x}-{1}+\frac{{1}}{{{x}-{1}}}+\frac{{1}}{{{x}+{2}}}$

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Friday, July 10, 2015

$\displaystyle\frac{{{{x}}^{{3}}-{x}+{3}}}{{{{x}}^{{2}}+{x}-{2}}}$ Write the partial fraction decomposition of the rational expression.

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