Problem 3 - Calculus - AP Edition

Question

Write out the form of the partial fraction decomposition. (Do not find the numerical values of the coefficients.) $$\frac{2 x-3}{x^{3}-x^{2}}$$

Step-by-Step Solution

Answer

Setup: \( \frac{2x-3}{x^3-x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} \).

Step 1: Factor the Denominator

First, factor the denominator of the given fraction \( x^3 - x^2 \). Notice that you can factor out \( x^2 \), thus the denominator becomes \( x^2(x - 1) \).

Step 2: Set Up Partial Fractions

Since the denominator \( x^2(x - 1) \) consists of a repeated linear factor \( x^2 \) and another linear factor \( x - 1 \), we set up the partial fraction decomposition as follows:\[ \frac{2x - 3}{x^3 - x^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} \]where \( A \), \( B \), and \( C \) are constants to be determined.

Key Concepts

Polynomial Fractions

Polynomial fractions, often referred to as rational expressions, are fractions where both the numerator and the denominator are polynomials. A typical example of a polynomial fraction might look like \(\frac{2x - 3}{x^3 - x^2}\). Such expressions are crucial in calculus and algebra as they appear frequently in various problems. To work effectively with polynomial fractions, it's important to simplify them whenever possible and understand their underlying structures, such as identifying common factors in the numerator and the denominator. This simplification process often involves techniques such as factoring, which helps in breaking down these expressions into more manageable pieces. By mastering polynomial fractions, students get better at handling more complex algebraic expressions.
Understanding these concepts can help simplify polynomial fractions, making them easier to work with in equations and calculus.

Factorization

Factorization is the process of breaking down a composite number or polynomial into a product of simpler factors. In the exercise provided, the denominator \(x^3 - x^2\) undergoes factorization to become \(x^2(x - 1)\). This process is pivotal because it transforms a complicated polynomial into an easily manageable form.
By factoring out common terms or identifying patterns such as the difference of squares or perfect square trinomials, one can simplify expressions significantly.

• Identifying common factors: In \(x^3 - x^2\), you notice \(x^2\) is common, hence it can be factored out.
• Resulting expression: After factorization, the expression simplifies to \(x^2(x-1)\).

This transformation lays the foundation for further simplification or decomposition in partial fractions, showcasing the power of factorization in algebra.

Linear Factors

Linear factors refer to expressions of the form \(ax + b\), where \(a\) and \(b\) are constants. In our example, the denominator \(x^2(x - 1)\) consists of linear factors: \(x\) and \(x-1\). Linear factors are crucial for setting up partial fractions because they simplify the decomposition process.
When you decompose the fraction \(\frac{2x - 3}{x^3 - x^2}\) into partial fractions, each term in the decomposition corresponds to a linear factor of the denominator. For instance, we denote them as \(\frac{A}{x}\), \(\frac{B}{x^2}\), and \(\frac{C}{x-1}\). Each of these denominators reflects a linear factor found in the factored form of \(x^3 - x^2\).
Understanding how to identify and use linear factors is essential for performing and simplifying partial fraction decompositions.