Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{4 x^{2}+3}{(x-5)^{3}} $$

Short answer

\[\frac{A}{x-5} + \frac{B}{(x-5)^{2}} + \frac{C}{(x-5)^{3}}\]

Step-by-step solution

Identify the structure of the denominator

The denominator of the given function is \((x-5)^{3}\), which is a cube of a linear term. In order to create the partial fractions, each term needs to be broken out separately. This means the decomposition will have terms related to \(x-5\), \((x-5)^{2}\), and \((x-5)^{3}\).

Write out the general form of the partial fraction decomposition

According to the rules of partial fractions, each term from the decomposition of the denominator in step 1 will have its own fraction in the decomposition of the function. The coefficients for these terms are what need to be found when solving for constants, but the problem states that we do not need to find them. So the general form would be: \[\frac{A}{x-5} + \frac{B}{(x-5)^{2}} + \frac{C}{(x-5)^{3}}\]where A, B, and C are constants that would need to be found if we were to fully decompose the function.