Problem 53 - College Algebra Essentials

Question

Explain what is meant by the partial fraction decomposition of a rational expression.

Step-by-Step Solution

Answer

The partial fraction decomposition of a rational expression is the process of expressing the rational expression as a sum of simpler fractions, with each of the fractions making up a part of the original expression. In essence, it breaks down a complex fraction into simpler, more manageable parts.

Step 1: Understanding Rational Expressions

A rational expression is a fraction where the numerator and the denominator are both polynomials. Essentially, it's a ratio of two polynomials and is analogous to a fraction. For instance, the expression \(\frac{x^2 + x - 2}{x - 1}\) is a rational expression where \(x^2 + x - 2\) is the numerator (a polynomial) and \(x - 1\) is the denominator (another polynomial).

Step 2: Concept of Partial Fractions

Partial fractions are the components of the rational expression that, when added together, form the original rational expression. That is, they represent the fraction in a simpler form where the denominator of each fraction is a factor of the original denominator. These fractions are 'partial' because each contains only part of the original fraction.

Step 3: Partial Fraction Decomposition

Partial fraction decomposition is the process of expressing the rational expression as the sum of simpler fractions, i.e., its partial fractions. The strategy is to rewrite the rational expression in a simpler form with the aid of its partial fractions. It's particularly useful for performing integrations, solving difference equations, and in other algebraic manipulations.

Step 4: Example

Let's consider a rational expression \(\frac{3x^2 - 2x - 1}{(x + 1)(x - 2)}\). To decompose it into partial fractions, fractions are set up with the denominators as the factors of the original denominator and unknown numerators. For instance, \(\frac{3x^2 - 2x - 1}{(x + 1)(x - 2)}\) is decomposed as \(\frac{A}{x + 1} + \frac{B}{x - 2}\), where A and B are values to be determined using algebraic methods to make both sides equal.