Blog: `(5 - x)/(2x^2 + x - 1)` Write the partial fraction decomposition of the rational expression.

Friday, July 25, 2008

$\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}$ Write the partial fraction decomposition of the rational expression.

$\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}$

$\displaystyle=\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{2}{x}-{x}-{1}}}$

$\displaystyle=\frac{{{5}-{x}}}{{{\left({2}{x}{\left({x}+{1}\right)}-{1}{\left({x}+{1}\right)}\right)}}}$

$\displaystyle=\frac{{{5}-{x}}}{{{\left({2}{x}-{1}\right)}{\left({x}+{1}\right)}}}$

Let $\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}=\frac{{A}}{{{2}{x}-{1}}}+\frac{{B}}{{{x}+{1}}}$

$\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}=\frac{{{A}{\left({x}+{1}\right)}+{B}{\left({2}{x}-{1}\right)}}}{{{\left({2}{x}-{1}\right)}{\left({x}+{1}\right)}}}$

$\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}=\frac{{{A}{x}+{A}+{2}{B}{x}-{B}}}{{{\left({2}{x}-{1}\right)}{\left({x}+{1}\right)}}}$

$\displaystyle\frac{{{5}-{x}}}{{{2}{{x}}^{{2}}+{x}-{1}}}=\frac{{{x}{\left({A}+{2}{B}\right)}+{A}-{B}}}{{{\left({2}{x}-{1}\right)}{\left({x}+{1}\right)}}}$