Problem 59 - 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. $$\frac{x-1}{x\left(x^{2}+1\right)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given expression is \[ \frac{A}{x} + \frac{B}{x^{2} + 1} + \frac{C}{(x^{2} + 1)^{2}} \]

Step 1: Identify the form of partial fraction

The first step is to identify the form of the partial fractions based on the denominator. Since there are three factors in the denominator: \( x \), \( x^{2}+1 \), and \( (x^{2}+1)^{2} \), the form of the fractions will be: \[ \frac{A}{x} + \frac{B}{x^{2} + 1} + \frac{C}{(x^{2} + 1)^{2}} \] for some constants A, B, and C.

Step 2: Identify the coefficients

We haven't been asked to solve for these constants in this exercise, we are only looking for the form of the decomposition. Therefore, the coefficients in this case remain undetermined and symbolized by A, B, and C.