Please help me solve with work shown ### Problem Statement**Question 22:** Write out the partial fraction decomposition for \(\frac{25x}{(x+2)(x-3)}\).### ExplanationPartial fraction decomposition is a method used to break down a complex rational expression into simpler fractions that are easier to work with, especially in integration. For the given rational function \(\frac{25x}{(x+2)(x-3)}\), we want to express it as a sum of simpler fractions. The denominator factors into linear terms, so we can decompose it into the form:\[\frac{25x}{(x+2)(x-3)} = \frac{A}{x+2} + \frac{B}{x-3}\]Here, \(A\) and \(B\) are constants that we need to determine.To find the values of \(A\) and \(B\), follow these steps:1. **Multiply both sides by the denominator \((x + 2)(x - 3)\):** \[ 25x = A(x - 3) + B(x + 2) \]2. **Expand and combine like terms:** \[ 25x = Ax - 3A + Bx + 2B \] \[ 25x = (A + B)x + (-3A + 2B) \]3. **Set up a system of equations by comparing the coefficients:** \[ A + B = 25 \] \[ -3A + 2B = 0 \]4. **Solve the system of equations to find \(A\) and \(B\):** From the second equation: \[ -3A + 2B = 0 \implies 2B = 3A \implies B = \frac{3}{2}A \] Substitute \(B = \frac{3}{2}A\) into the first equation: \[ A + \frac{3}{2}A = 25 \implies \frac{5}{2}A = 25 \implies A = 10 \] Then, \[ B = \frac{3}{2}A \implies B = \frac{

Name: Please help me solve with work shown ### Problem Statement**Question 2
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please help me solve with work shown ### Problem Statement**Question 22:** Write out the partial fraction decomposition for \(\frac{25x}{(x+2)(x-3)}\).### ExplanationPartial fraction decomposition is a method used to break down a complex rational exp