Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$ \frac{2 x-3}{x^{3}+10 x} $$

Short answer

\( \frac{2x-3}{x(x^{2}+10)} = \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \)

Step-by-step solution

Factorize the Denominator Polynomial

To begin with, the denominator polynomial of the rational expression needs to be factored. For the expression \( \frac{2x-3}{x^{3}+10x} \), the denominator \( x^{3}+10x \) can be factorized as \( x(x^{2}+10) \).

Formulate the Partial Fraction Decomposition

Following the factorisation, the partial fraction decomposition of the rational expression can be presented with unknown constants. Since the denominator \( x(x^{2}+10) \) has a linear term \( x \), and a quadratic term \( x^{2}+10 \), the partial fraction decomposition will include two fractions, each corresponding to these factors, i.e: \( \frac{2x-3}{x(x^{2}+10)} = \frac{A}{x} + \frac{Bx + C}{x^{2}+10} \). As we are not required to find the values of the constants \( A \), \( B \), and \( C \), this is the final answer.