Blog: `(3x^2 - 7x - 2)/(x^3 - x)` Write the partial fraction decomposition of the rational expression.

Tuesday, March 30, 2010

$\displaystyle\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{{x}}^{{3}}-{x}}}$ Write the partial fraction decomposition of the rational expression.

$\displaystyle\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{{x}}^{{3}}-{x}}}=\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{x}{\left({x}+{1}\right)}{\left({x}-{1}\right)}}}$

$\displaystyle\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{{x}}^{{3}}-{x}}}=\frac{{A}}{{x}}+\frac{{B}}{{{x}+{1}}}+\frac{{C}}{{{x}-{1}}}$

Multiply through by the LDC $\displaystyle{{x}}^{{3}}-{x}.$

$\displaystyle{3}{{x}}^{{2}}-{7}{x}-{2}={A}{\left({{x}}^{{2}}-{1}\right)}+{B}{x}{\left({x}-{1}\right)}+{C}{x}{\left({x}+{1}\right)}$

$\displaystyle{3}{{x}}^{{2}}-{7}{x}-{2}={A}{{x}}^{{2}}-{A}+{B}{{x}}^{{2}}-{B}{x}+{C}{{x}}^{{2}}+{C}{x}$

$\displaystyle{3}{{x}}^{{2}}-{7}{x}-{2}={\left({A}+{B}+{C}\right)}{{x}}^{{2}}+{\left(-{B}+{C}\right)}{x}-{A}$

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

$\displaystyle{3}={A}+{B}+{C}$

$\displaystyle-{7}=-{B}+{C}$

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

$\displaystyle{A}={2}$

$\displaystyle{3}={A}+{B}+{C}$

$\displaystyle{3}={2}+{B}+{C}$

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

Use the elimination method to solve for B and C.

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

$\displaystyle-{7}=-{B}+{C}$

_______________

$\displaystyle-{6}={2}{C}$

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

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

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

$\displaystyle{4}={B}$

$\displaystyle{A}={2},{B}={4},{C}=-{3}$

$\displaystyle\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{{x}}^{{3}}-{x}}}=\frac{{2}}{{x}}+\frac{{4}}{{{x}+{1}}}-\frac{{3}}{{{x}-{1}}}$

Blog

Tuesday, March 30, 2010

$\displaystyle\frac{{{3}{{x}}^{{2}}-{7}{x}-{2}}}{{{{x}}^{{3}}-{x}}}$ Write the partial fraction decomposition of the rational expression.

No comments:

Post a Comment

How does author Elie Wiesel use symbolism to contribute to the meaning of Night?