In Exercises 9–42, write the partial fraction decomposition of each rational expression. 9x+21/(x² + 2x - 15)

Verified step by step guidance

1. Step 1: Factor the denominator. The expression in the denominator is $x^2+2x-15$. Find two numbers that multiply to -15 and add to 2.
2. Step 2: Once you have the factors, rewrite the denominator as a product of two binomials.
3. Step 3: Set up the partial fraction decomposition. Since the denominator is now a product of two linear factors, express the fraction as $\frac{A}{\text{first factor}}+\frac{B}{\text{second factor}}$.
4. Step 4: Clear the fractions by multiplying through by the common denominator to obtain an equation without fractions.
5. Step 5: Solve for the constants $A$ and $B$ by equating coefficients or substituting convenient values for $x$.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial for performing operations such as addition, subtraction, and decomposition. In this case, the expression 9x + 21/(x² + 2x - 15) is a rational expression that needs to be decomposed into simpler fractions.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions. This technique is particularly useful for integrating rational functions or simplifying complex expressions. The goal is to break down the given rational expression into components that are easier to work with, based on the factors of the denominator.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This is essential in partial fraction decomposition, as the first step is to factor the denominator completely. For the expression x² + 2x - 15, identifying its factors will allow us to set up the correct form for the partial fractions.