Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-1}{x\left(x^{2}+1\right)^{2}} $$

Short answer

The partial fraction decomposition of the given rational expression, without solving for the coefficients, can be written as \(\frac{a}{x} + \frac{bx + c}{x^2+1} + \frac{dx + e}{(x^2+1)^2}\).

Step-by-step solution

Identifying the General Form

For the given rational expression, \(\frac{2x - 1}{x(x^2+1)^2}\), we first need to identify the general form of the partial fractions decomposition. The distinct factors are \(x\) and \((x^2 + 1)\). Since \((x^2 + 1)\) appears twice in the denominator, we account for it twice in the decomposition. Therefore, our general form will look like: A/x + B/(x^2+1) + C/(x^2+1)^2

Expressing the Coefficients as Variables

Next, to accommodate the degree of \(x\) in the numerator, we generalize the numerators of the fractions with polynomial variables. We shift from constants (\(A, B, C\)) to variables (\(a, bx+c, dx+e\)). Now the desired form of partial fraction decomposition for the given equation will be:\n\[\frac{a}{x} + \frac{bx + c}{x^2+1} + \frac{dx + e}{(x^2+1)^2}\]