Blog: `(6x^2 + 1)/(x^2(x - 1)^2)` Write the partial fraction decomposition of the rational expression. Check your result algebraically.

Tuesday, November 20, 2012

$\displaystyle\frac{{{6}{{x}}^{{2}}+{1}}}{{{{x}}^{{2}}{{\left({x}-{1}\right)}}^{{2}}}}$ Write the partial fraction decomposition of the rational expression. Check your result algebraically.

$\displaystyle\frac{{{6}{{x}}^{{2}}+{1}}}{{{{x}}^{{2}}{{\left({x}-{1}\right)}}^{{2}}}}=\frac{{A}}{{x}}+\frac{{B}}{{{x}}^{{2}}}+\frac{{C}}{{{x}-{1}}}+\frac{{D}}{{{\left({x}-{1}\right)}}^{{2}}}$

Multiply by the LCD $\displaystyle{{x}}^{{2}}{{\left({x}-{1}\right)}}^{{2}}.$

$\displaystyle{6}{{x}}^{{2}}+{1}={A}{x}{{\left({x}-{1}\right)}}^{{2}}+{B}{{\left({x}-{1}\right)}}^{{2}}+{C}{{x}}^{{2}}{\left({x}-{1}\right)}+{D}{{x}}^{{2}}$

$\displaystyle{6}{{x}}^{{2}}+{1}={A}{x}{\left({{x}}^{{2}}-{2}{x}+{1}\right)}+{B}{\left({{x}}^{{2}}-{2}{x}+{1}\right)}+{C}{{x}}^{{3}}-{C}{{x}}^{{2}}+{D}{{x}}^{{2}}$

$\displaystyle{6}{{x}}^{{2}}+{1}={A}{{x}}^{{3}}-{2}{A}{{x}}^{{2}}+{A}{x}+{B}{{x}}^{{2}}-{2}{B}{x}+{B}+{C}{{x}}^{{3}}-{C}{{x}}^{{2}}+{D}{{x}}^{{2}}$

$\displaystyle{6}{{x}}^{{2}}+{1}={\left({A}+{C}\right)}{{x}}^{{3}}+{\left(-{2}{A}+{B}-{C}+{D}\right)}{{x}}^{{2}}+{\left({A}-{2}{B}\right)}{x}+{B}$

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

B=1

$\displaystyle{0}={A}-{2}{B}$

$\displaystyle{0}={A}-{2}{\left({1}\right)}$

$\displaystyle{A}={2}$

$\displaystyle{0}={A}+{C}$

$\displaystyle{0}={2}+{C}$

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

$\displaystyle{6}=-{2}{A}+{B}-{C}+{D}$

$\displaystyle{6}=-{2}{\left({2}\right)}+{1}-{\left(-{2}\right)}+{D}$

$\displaystyle{6}=-{4}+{1}+{2}+{D}$

$\displaystyle{D}={7}$

$\displaystyle{A}={2},{B}={1},{C}=-{2},{D}={7}$

$\displaystyle\frac{{{6}{{x}}^{{2}}+{1}}}{{{{x}}^{{2}}{{\left({x}-{1}\right)}}^{{2}}}}=\frac{{2}}{{x}}+\frac{{1}}{{{x}}^{{2}}}+{\left[-\frac{{2}}{{{x}-{1}}}\right]}+\frac{{7}}{{{\left({x}-{1}\right)}}^{{2}}}$

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Tuesday, November 20, 2012

$\displaystyle\frac{{{6}{{x}}^{{2}}+{1}}}{{{{x}}^{{2}}{{\left({x}-{1}\right)}}^{{2}}}}$ Write the partial fraction decomposition of the rational expression. Check your result algebraically.

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