Blog: `(x^2 + 12x + 12)/(x^3 - 4x)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

Wednesday, November 3, 2010

$\displaystyle\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{{x}}^{{3}}-{4}{x}}}$ Write the partial fraction decomposition of the rational expression. Check your result algebraically.

$\displaystyle\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{{x}}^{{3}}-{4}{x}}}=\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{x}{\left({x}+{2}\right)}{\left({x}-{2}\right)}}}$

$\displaystyle\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{{x}}^{{3}}-{4}{x}}}=\frac{{A}}{{x}}+\frac{{B}}{{{x}+{2}}}+\frac{{C}}{{{x}-{2}}}$

Multiply through by the LCD $\displaystyle{{x}}^{{3}}+{4}{x}.$

$\displaystyle{{x}}^{{2}}+{12}{x}+{12}={A}{\left({{x}}^{{2}}-{4}\right)}+{B}{x}{\left({x}-{2}\right)}+{C}{x}{\left({x}+{2}\right)}$

$\displaystyle{{x}}^{{2}}+{12}{x}+{12}={A}{{x}}^{{2}}-{4}{A}+{B}{{x}}^{{2}}-{2}{B}{x}+{C}{{x}}^{{2}}+{2}{C}{x}$

$\displaystyle{{x}}^{{2}}+{12}{x}+{12}={\left({A}+{B}+{C}\right)}{{x}}^{{2}}+{\left(-{2}{B}+{2}{C}\right)}{x}+{\left(-{4}{A}\right)}$

Equate coefficient of like terms. Then solve for A, B, and C.

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

$\displaystyle{12}=-{2}{B}+{2}{C}$

$\displaystyle{12}=-{4}{A}$

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

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

$\displaystyle{1}=-{3}+{B}+{C}$

$\displaystyle{4}={B}+{C}$

Solve for B and C using the elimination method.

$\displaystyle{4}={B}+{C}$

$\displaystyle{12}=-{2}{B}+{2}{C}$

Multiply the first equation by 2. Then solve using the elimination method.

$\displaystyle{8}={2}{B}+{2}{C}$

$\displaystyle{12}=-{2}{B}+{2}{C}$

_________________

20=4C

$\displaystyle{C}={5}$

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

$\displaystyle{1}=-{3}+{B}+{5}$

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

$\displaystyle{A}=-{3},{B}=-{1},{C}={5}$

$\displaystyle\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{{x}}^{{3}}-{4}{x}}}=-\frac{{3}}{{x}}-\frac{{1}}{{{x}+{2}}}+\frac{{5}}{{{x}-{2}}}$

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Wednesday, November 3, 2010

$\displaystyle\frac{{{{x}}^{{2}}+{12}{x}+{12}}}{{{{x}}^{{3}}-{4}{x}}}$ Write the partial fraction decomposition of the rational expression. Check your result algebraically.

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