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Chapter 8: Problem 57

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

Expert verified
The form of the partial fraction decomposition of the given rational expression is \[ \frac{A}{(x-5)} + \frac{B}{(x-5)^2} + \frac{C}{(x-5)^3} \]

Step by step solution

01

Identify the form of the denominator

The denominator of the rational expression is \((x-5)^{3}\). This means it has one distinct factor, which is \(x-5\), but it has a multiplicity of 3 as it is raised to the power of 3.
02

Write the form of the partial fraction decomposition based on the denominator

Based on the information from the previous step, the form of the partial fraction decomposition can be written as: 'A' over \((x-5)\), 'B' over \((x-5)^2\), and 'C' over \((x-5)^3\), since the denominator factor \(x-5\) has a multiplicity of 3. The whole form becomes: \[ \frac{A}{(x-5)} + \frac{B}{(x-5)^2} + \frac{C}{(x-5)^3} \].

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