Problem 58 - 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. $$\frac{6 x+5}{(x+2)^{4}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \( \frac{A}{x+2} + \frac{B}{(x+2)^2}+ \frac{C}{(x+2)^3}+ \frac{D}{(x+2)^4} \), where A, B, C, and D are constants.

Step 1: Identify the highest power in the denominator

The denominator of the rational expression is \( (x+2)^{4} \). This can be written as 4 terms where the power of x ranges from 3 (the highest) to 0.

Step 2: Write down the Partial Fraction Decomposition form

The form of the partial fraction decomposition for the given problem, using distinct denominators for each term and constants for the numerators, can be written as follows: \( \frac{A}{x+2} + \frac{B}{(x+2)^2}+ \frac{C}{(x+2)^3}+ \frac{D}{(x+2)^4} \), where A, B, C, and D are constants to be determined if we were to solve the actual decomposition. Since the task is to only write the form, we stop at this step.