- Factor the denominator: \(x^2-2x-15 = (x+3)(x-5)\).
- Decompose the expression \(\frac{2x+3}{(x+3)(x-5)}\) using partial fractions, yielding \(\frac{3/8}{x+3} + \frac{13/8}{x-5}\).
- Scale the fractions to match the required denominator format: \(\frac{\frac{3x-15}{8}}{(x+3)(x-5)} + \frac{\frac{13x+39}{8}}{(x-5)(x+3)}\).
- Final Result: \(\frac{2x+3}{x^2-2x-15} = \boxed{\frac{\frac{3}{8}x - \frac{15}{8}}{(x+3)(x-5)} + \frac{\frac{13}{8}x + \frac{39}{8}}{(x-5)(x+3)}}\)
Explanation
- Analysis and Decomposition Setup
The objective is to express the rational expression \(\frac{2x+3}{x^2-2x-15}\) as a sum of two rational expressions with the denominator \((x+3)(x-5)\). First, we factor the denominator: \(x^2-2x-15 = (x+3)(x-5)\). We perform a partial fraction decomposition by setting \(\frac{2x+3}{(x+3)(x-5)} = \frac{A}{x+3} + \frac{B}{x-5}\). Multiplying by \((x+3)(x-5)\) gives \(2x+3 = A(x-5) + B(x+3)\).
- Finding Constants A and B
To find the constants \(A\) and \(B\), we substitute specific values for \(x\). If \(x=5\), then \(2(5)+3 = B(5+3)\), which simplifies to \(13 = 8B\), so \(B = \frac{13}{8}\). If \(x=-3\), then \(2(-3)+3 = A(-3-5)\), which simplifies to \(-3 = -8A\), so \(A = \frac{3}{8}\). Thus, the partial fraction decomposition is \(\frac{3/8}{x+3} + \frac{13/8}{x-5}\).
- Formatting the Final Answer
We rewrite the partial fractions to have a common denominator \((x+3)(x-5)\). For the first term, we multiply the numerator and denominator by \((x-5)\) to get \(\frac{3/8(x-5)}{(x+3)(x-5)} = \frac{3/8x - 15/8}{(x+3)(x-5)}\). For the second term, we multiply the numerator and denominator by \((x+3)\) to get \(\frac{13/8(x+3)}{(x-5)(x+3)} = \frac{13/8x + 39/8}{(x-5)(x+3)}\). Adding these, we get \(\frac{3/8x - 15/8 + 13/8x + 39/8}{(x+3)(x-5)} = \frac{2x+3}{(x+3)(x-5)}\), confirming the decomposition.
Examples
Partial fraction decomposition is essential in calculus, especially when computing integrals of complex rational functions. For example, if an engineer needs to integrate \(\int \frac{2x+3}{x^2-2x-15} dx\), they would use this technique to transform the integrand into \(\int (\frac{3/8}{x+3} + \frac{13/8}{x-5}) dx\), which is much easier to evaluate as \(\frac{3}{8}\ln|x+3| + \frac{13}{8}\ln|x-5| + C\). This method is widely applied in fields like signal processing, control systems, and thermodynamics to simplify complex system behavior models.