Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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 1: 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.

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