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

Verified step by step guidance

1. Step 1: Identify the degree of the numerator and the denominator. The degree of both the numerator and the denominator is 2. If the degree of the numerator is greater than or equal to the degree of the denominator, we would need to perform polynomial division first. However, in this case, the degrees are equal, so we can proceed to the next step.
2. Step 2: Factor the denominator. The denominator x^2 + 2x + 1 can be factored into (x+1)^2.
3. Step 3: Write the expression as a sum of partial fractions. Since the denominator is a repeated linear factor (x+1)^2, we write the expression as A/(x+1) + B/(x+1)^2.
4. Step 4: Multiply through by the common denominator to clear the fractions. This gives us x^2 = A*(x+1) + B.
5. Step 5: Solve for A and B by substituting convenient values for x and comparing coefficients on both sides of the equation.

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 case, the expression (x^2)/(x^2 + 2x + 1) is a rational expression that needs to be analyzed for its partial fraction decomposition.

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 simpler components that can be more easily manipulated or integrated.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can be linear or irreducible quadratic expressions. In the context of partial fraction decomposition, factoring the denominator is essential to identify the appropriate form of the partial fractions. For the expression (x^2 + 2x + 1), recognizing it as (x + 1)^2 is key to proceeding with the decomposition.