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

Friday, March 21, 2014

$\displaystyle\frac{{{3}{x}}}{{{\left({x}-{3}\right)}}^{{2}}}$ Write the partial fraction decomposition of the rational expression. Check your result algebraically.

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

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

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

$\displaystyle\therefore{3}{x}={A}{\left({x}-{3}\right)}+{B}$

$\displaystyle{3}{x}={A}{x}-{3}{A}+{B}$

equating the coefficients of the like terms,

$\displaystyle{A}={3}$

$\displaystyle-{3}{A}+{B}={0}$

plug the value of the A in the above equation,

$\displaystyle-{3}{\left({3}\right)}+{B}={0}$

$\displaystyle-{9}+{B}={0}$

$\displaystyle{B}={9}$

$\displaystyle\therefore\frac{{{3}{x}}}{{{\left({x}-{3}\right)}}^{{2}}}=\frac{{3}}{{{x}-{3}}}+\frac{{9}}{{{\left({x}-{3}\right)}}^{{2}}}$

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

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

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

$\displaystyle\therefore{3}{x}={A}{\left({x}-{3}\right)}+{B}$

$\displaystyle{3}{x}={A}{x}-{3}{A}+{B}$

equating the coefficients of the like terms,

$\displaystyle{A}={3}$

$\displaystyle-{3}{A}+{B}={0}$

plug the value of the A in the above equation,

$\displaystyle-{3}{\left({3}\right)}+{B}={0}$

$\displaystyle-{9}+{B}={0}$

$\displaystyle{B}={9}$

$\displaystyle\therefore\frac{{{3}{x}}}{{{\left({x}-{3}\right)}}^{{2}}}=\frac{{3}}{{{x}-{3}}}+\frac{{9}}{{{\left({x}-{3}\right)}}^{{2}}}$

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)