In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^3 + 5x^2 + 7x - 1)/(x^2 + x + 1)^2

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1. Step 1: Recognize that the given rational expression is a proper fraction, as the degree of the numerator (4x^3 + 5x^2 + 7x - 1) is less than the degree of the denominator ((x^2 + x + 1)^2). This means partial fraction decomposition is applicable.
2. Step 2: Factor the denominator if possible. In this case, the denominator is already expressed as (x^2 + x + 1)^2, which is a repeated irreducible quadratic factor.
3. Step 3: Set up the partial fraction decomposition. For a repeated irreducible quadratic factor like (x^2 + x + 1)^2, the decomposition will take the form: A(x) / (x^2 + x + 1) + B(x) / (x^2 + x + 1)^2, where A(x) and B(x) are polynomials of degree less than the degree of the quadratic factor (degree < 2). Thus, A(x) = Ax + B and B(x) = Cx + D.
4. Step 4: Write the equation for the decomposition: (4x^3 + 5x^2 + 7x - 1) / (x^2 + x + 1)^2 = (Ax + B) / (x^2 + x + 1) + (Cx + D) / (x^2 + x + 1)^2.
5. Step 5: Multiply through by the denominator (x^2 + x + 1)^2 to eliminate the fractions, resulting in: 4x^3 + 5x^2 + 7x - 1 = (Ax + B)(x^2 + x + 1) + (Cx + D). Expand and collect like terms, then equate coefficients of corresponding powers of x to solve for A, B, C, and D.

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, multiplication, and division, as well as for decomposing them into simpler components, which is the focus of the given question.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions and involves breaking down the expression based on the factors of the denominator, allowing for easier manipulation and analysis of the function.

Polynomial Long Division

Polynomial long division is a method 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, as it simplifies the rational expression before applying partial fraction decomposition.