Find the partial fraction decomposition for each rational expression. See Examples 1–4. (2x^5 + 3x^4 - 3x^3 - 2x^2 + x)/(2x^2 + 5x + 2)

Verified step by step guidance

1. Step 1: Check if the degree of the numerator is greater than the degree of the denominator. If it is, perform polynomial division to reduce the degree of the numerator. In this case, the degree of the numerator (5) is greater than the degree of the denominator (2), so we need to perform polynomial division.
2. Step 2: After performing the polynomial division, you will get a quotient and a remainder. The quotient will be a polynomial, and the remainder will be a polynomial of degree less than the denominator.
3. Step 3: Write the original rational expression as the sum of the quotient and the remainder divided by the denominator. This is the first step in the partial fraction decomposition.
4. Step 4: Now, factor the denominator of the fraction obtained in step 3. In this case, the denominator is a quadratic polynomial, so you can factor it into two linear factors.
5. Step 5: Write the fraction as a sum of fractions, where each fraction has one of the factors of the denominator as its denominator. The coefficients of these fractions are unknowns that you will need to solve for. Set up a system of equations by equating the coefficients of like terms on both sides of the equation, and solve this system to find the unknowns. These unknowns are the coefficients of the partial fractions.

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^5 + 3x^4 - 3x^3 - 2x^2 + x)/(2x^2 + 5x + 2) 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 original rational expression into components that are easier to work with, typically involving linear or irreducible quadratic factors in the denominator.

Polynomial Long Division

Polynomial long division is a process used to divide one polynomial by another, similar to numerical long division. This technique 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, performing polynomial long division may be necessary to simplify the rational expression before applying the decomposition method.