Blog: `(x^4 + 2x^3 + 4x^2 + 8x + 2)/(x^3 + 2x^2 + x)` Write the partial fraction decomposition of the improper rational expression.

Friday, August 21, 2015

$\displaystyle\frac{{{{x}}^{{4}}+{2}{{x}}^{{3}}+{4}{{x}}^{{2}}+{8}{x}+{2}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}$ Write the partial fraction decomposition of the improper rational expression.

A bit more extension for my above solution

Now in the above expression we need to simplify the

$\displaystyle\frac{{{2}{x}-{1}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}$
It is as follows

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

$\displaystyle\frac{{{2}{x}-{1}}}{{{x}{{\left({x}+{1}\right)}}^{{2}}}}={\left(\frac{{a}}{{x}}\right)}+{\left(\frac{{b}}{{{x}+{1}}}\right)}+{\left(\frac{{c}}{{{\left({x}+{1}\right)}}^{{2}}}\right)}{\)}$

on simplification we get
$\displaystyle{\left({2}{x}-{1}\right)}={\left({a}{{\left({x}+{1}\right)}}^{{2}}\right)}+{\left({b}{x}{\left({x}+{1}\right)}\right)}+{c}{x}$

As the roots of the denominator $\displaystyle{\left({x}{{\left({x}+{1}\right)}}^{{2}}\right)}$ are $\displaystyle{0},-{1}$ . We can solve the unknown parameters by
plugging the values of $\displaystyle{x}$ .

when $\displaystyle{x}={0}$ , we get
$\displaystyle{a}=-{1}$
when $\displaystyle{x}={\left(-{1}\right)}$ we get
$\displaystyle{c}={3}$

As we know the $\displaystyle{a},{c}$ values , we can find the value of $\displaystyle{b}$ as

$\displaystyle{2}{x}-{1}={\left(-{1}\right)}{{\left({x}+{1}\right)}}^{{2}}+{b}{x}{\left({x}+{1}\right)}+{3}{x}$
$\displaystyle{2}{x}-{1}={b}{{x}}^{{2}}+{x}+{b}{x}-{{x}}^{{2}}-{1}$
$\displaystyle{2}{x}-{1}={{x}}^{{2}}{\left({b}-{1}\right)}+{x}{\left({b}+{1}\right)}-{1}$
on comparing we get
$\displaystyle{b}+{1}={2}$
=> $\displaystyle{b}={1}$
so, $\displaystyle\frac{{{2}{x}-{1}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}={\left(\frac{{-{1}}}{{x}}\right)}+{\left(\frac{{1}}{{{x}+{1}}}\right)}+{\left(\frac{{3}}{{{x}+{{1}}^{{2}}}}\right)}$
so, the partial fraction for

$\displaystyle\frac{{{{x}}^{{4}}+{2}{{x}}^{{3}}+{4}{{x}}^{{2}}+{8}{x}+{2}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}={x}+{\left(\frac{{3}}{{x}}\right)}+\frac{{{2}{x}-{1}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}$

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

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Friday, August 21, 2015

$\displaystyle\frac{{{{x}}^{{4}}+{2}{{x}}^{{3}}+{4}{{x}}^{{2}}+{8}{x}+{2}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}$ Write the partial fraction decomposition of the improper rational expression.

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What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?

Friday, August 21, 2015

Write the partial fraction decomposition of the improper rational expression.

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What is the Exposition, Rising Action, Climax, and Falling Action of &quot;One Thousand Dollars&quot;?

$\displaystyle\frac{{{{x}}^{{4}}+{2}{{x}}^{{3}}+{4}{{x}}^{{2}}+{8}{x}+{2}}}{{{{x}}^{{3}}+{2}{{x}}^{{2}}+{x}}}$ Write the partial fraction decomposition of the improper rational expression.

What is the Exposition, Rising Action, Climax, and Falling Action of "One Thousand Dollars"?