Problem 3 - Precalculus with Calculus Previews

Question

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{2 x^{2}-3}{x^{3}+x^{2}} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1}\).

Step 1: Factor the Denominator

To decompose the rational expression into partial fractions, first, factor the denominator \(x^3 + x^2\). The expression can be factored as follows: \(x^3 + x^2 = x^2(x + 1)\).

Step 2: Set Up Partial Fractions

With the factored form of the denominator \(x^2(x + 1)\), we set up the partial fraction decomposition. The decomposition takes the form: \[\frac{2x^2 - 3}{x^2(x + 1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 1}\] where \(A\), \(B\), and \(C\) are constants to be determined.

Key Concepts

Rational Expression

Rational expressions are a key concept in algebra. Similar to fractions, a rational expression is defined as a ratio of two polynomials. The expression involves a numerator and a denominator. For the given exercise, the rational expression is \( \frac{2x^2 - 3}{x^3 + x^2} \). Notice how both the numerator (\(2x^2 - 3\)) and the denominator (\(x^3 + x^2\)) are polynomial expressions. Understanding rational expressions is crucial because they often arise in calculus and algebra problems.
Due to the presence of variables in the denominator, we need to handle these expressions carefully to avoid division by zero. The aim is to simplify these expressions through methods like partial fraction decomposition, which is a technique used to break complex rational expressions into simpler fractions.

Factorization

Factorization is a fundamental process in simplifying rational expressions. It involves expressing a polynomial as a product of its factors, which are polynomials of lower degree. This process is crucial because many algebraic techniques, like partial fraction decomposition, depend on factorization.
In the exercise, the denominator \( x^3 + x^2 \) is factored to \( x^2(x + 1) \). To factor this, we look for common terms in the polynomial. Notice that \( x^2 \) is a common factor, which can be factored out to simplify the expression.
Understanding how to factor polynomials properly is vital as it reveals the structure needed to set up partial fraction decomposition in our rational expression.

Constants Determination

After factorizing the denominator, the next step in partial fraction decomposition is setting up the expression to determine constants (often represented as A, B, C, etc.). These constants are crucial as they fully define each term in the simplified expression.
Consider the partial fraction setup for this exercise:

• \( \frac{A}{x} \) expresses the term dealing with the single power of \( x \).
• \( \frac{B}{x^2} \) handles the higher power of \( x \).
• \( \frac{C}{x + 1} \) deals with the linear polynomial \( x + 1 \).

The goal is to express the rational expression \( \frac{2x^2 - 3}{x^2(x + 1)} \) as a sum of these simpler fractions, which makes it easier to integrate or simplify.
Note that in many exercises, the actual determination of these constants requires solving for them using polynomial identities or equations. In this exercise, however, we are only setting up the expression, leaving the determination of constants out of the scope.