Calculation:

Consider the provided expression:

\(\displaystyle{\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}^{{{3}}}+{7}{x}}}}\)

Here, \(\displaystyle{f{{\left({x}\right)}}}={x}?-{2}{x}-{21}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={x}^{{\circ}}+{T}{x}\)

Factorize \(\displaystyle{g}{\left({x}\right)}:\)

\(\displaystyle{g}{\left({x}\right)}={x}^{{{3}}}+{7}{x}\)

\(\displaystyle={x}{\left({x}^{{{2}}}+{7}\right)}\)

So, the expression can be written as:

\(\displaystyle{\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}^{{{3}}}+{7}{x}}}}={\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}{\left({x}^{{{3}}}+{7}{x}\right)}}}}\)

Here, in g (x), there are two factors, one linear and one quadratic. So, the expression can be decomposed as:

\(\displaystyle{\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}{\left({x}^{{{3}}}+{7}{x}\right)}}}}={\frac{{{A}}}{{{x}}}}+{\frac{{{B}{x}+{C}}}{{{x}^{{{2}}}+{7}}}}\)

Here, Least Common Divisor is \(\displaystyle{x}{\left({x}^{{{2}}}+{7}\right)}\)

Multiply both sides of (1) by the Least Common Divisor to clear fractions:

\(\displaystyle{x}{\left({x}^{{{2}}}+{7}\right)}{\left[{\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}{\left({x}^{{{2}}}+{7}{x}\right)}}}}\right]}={x}{\left({x}^{{{2}}}+{7}\right)}{\left[{\frac{{-{A}}}{{{x}}}}+{\frac{{{B}{x}+{C}}}{{{x}^{{{2}}}+{7}{x}}}}\right]}\)

simplify to obtain:

\(\displaystyle{x}^{{{2}}}+{2}{x}-{21}={A}{\left({x}^{{{2}}}+{7}\right)}+{\left({B}{x}+{C}\right)}{x}\)

\(\displaystyle{x}^{{{2}}}+{2}{x}-{21}={A}{x}^{{{2}}}+{7}+{B}{x}^{{{2}}}+{C}{x}\)

\(\displaystyle{x}^{{{2}}}+{2}{x}-{21}={\left({A}+{C}\right)}{x}^{{{2}}}+{C}{x}+{7}{A}\)

Compare the coefficients of x,x? and constant terms:

\(\displaystyle{A}+{B}={1},{C}=-{2}\text{nd7A=-2}{n}{d}{7}{A}=-{21}\)

Solve for A the equation \(\displaystyle{7}{A}=—{21}\)

\(\displaystyle{7}{A}=-{21}\)

\(\displaystyle{A}=-{3}\)

Substitute -3 for A in \(\displaystyle{A}+{B}={1}\) and simplify for B:

\(\displaystyle{\left(-{3}\right)}+{B}={1}\)

\(\displaystyle{B}={4}\)

Substitute the obtained values of A, B and Cin (1):

\(\displaystyle{\frac{{{x}^{{{2}}}-{2}{x}-{21}}}{{{x}{\left({x}^{{{3}}}+{7}{x}\right)}}}}={\frac{{{3}}}{{{x}}}}+{\frac{{{4}{x}-{2}}}{{{x}^{{{2}}}+{7}}}}\)

Therefore, the partial fraction decomposition for \(\frac{x^{2}-2x-21}{x\left(x^{3}+7x\right)}\text is\frac{3}{x}+\frac{4x-2}{x^{2}+7}\)