Problem 54 - 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. $$\frac{x-2}{x^{2}+4 x+3}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \(\frac{x-2}{x^{2}+4 x+3}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x+3}\)

Step 1: Factoring the denominator

Factor the quadratic expression in the denominator. \(x^{2}+4 x+3\) can be factored into \((x+1)(x+3)\)

Step 2: Set up the partial fraction decomposition

Set up the partial fraction decomposition. The general form of the decomposition will be: \(\frac{x-2}{x^{2}+4 x+3}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x+3}\) where A and B are constants to be determined.