Answer
The stated partial fraction decomposition is incorrect.
Work Step by Step
It is provided that $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{\left( x-4 \right)}$ is the partial fraction decomposition of the algebraic expression $\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ into a series of smaller components.
We know that the partial fraction decomposition is the technique used for rational functions. It is the mathematical operation by which rational function can be expressed as a sum of a polynomial and several fractions, including ‘equating coefficients’.
And the partial fraction decomposition associated with $\frac{p{{x}^{2}}+qx+r}{\left( x-a \right){{\left( x-b \right)}^{2}}},a\ne b$, is:
$\frac{A}{\left( x-a \right)}+\frac{B}{\left( x-b \right)}+\frac{C}{{{\left( x-b \right)}^{2}}}$.
Therefore, $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{{{\left( x-4 \right)}^{2}}}$ is the final simplified algebraic expression
Thus, the stated partial fraction decomposition $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{\left( x-4 \right)}$ of $\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ is incorrect and the correct partial fraction decomposition is as follows:
$\frac{3x}{\left( x+5 \right){{\left( x-4 \right)}^{2}}}$ $=$ $\frac{A}{\left( x+5 \right)}+\frac{B}{\left( x-4 \right)}+\frac{C}{{{\left( x-4 \right)}^{2}}}$ .
