Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x + 1)/(x + 2)^3

Verified step by step guidance

1. Identify the form of the partial fraction decomposition. Since the denominator is $(x+2{)}^3$, the decomposition will have terms of the form $\frac{A}{x+2}+\frac{B}{(x+2{)}^2}+\frac{C}{(x+2{)}^3}$.
2. Set up the equation: $\frac{2x+1}{(x+2{)}^3}=\frac{A}{x+2}+\frac{B}{(x+2{)}^2}+\frac{C}{(x+2{)}^3}$.
3. Multiply through by $(x+2{)}^3$ to clear the denominators: $2x+1=A(x+2{)}^2+B(x+2)+C$.
4. Expand $A(x+2{)}^2$ and $B(x+2)$ to simplify the equation: $A(x^2+4x+4)+B(x+2)+C$.
5. Combine like terms and equate coefficients with $2x+1$ to solve for $A$, $B$, and $C$.

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 like addition, subtraction, and decomposition. In this context, the expression (2x + 1)/(x + 2)^3 is a rational expression that requires decomposition 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 process involves breaking down the rational expression based on the factors of the denominator, which in this case is (x + 2)^3.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It is essential when the degree of the numerator is greater than or equal to the degree of the denominator. In the context of partial fraction decomposition, if the rational expression is improper, polynomial long division may be necessary before applying the decomposition technique.