Problem 3 - Precalculus: Mathematics for Calculus

Question

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{1}{(x-1)(x+2)}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Denominator

Examine the rational function \( \frac{1}{(x-1)(x+2)} \). The denominator is \((x-1)(x+2)\), which indicates the factors involved in the partial fraction decomposition.

Step 2: Set Up the Partial Fraction Form

The partial fraction decomposition involves expressing \( \frac{1}{(x-1)(x+2)} \) as the sum of fractions where each factor of the denominator appears in one fraction. Thus, it would be: \[ \frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} \] where \(A\) and \(B\) are constants (coefficients) to be determined.

Key Concepts

Rational Functions

A rational function is any function that can be written as the ratio of two polynomials. Specifically, the general form is given by \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials, and the denominator \(Q(x)\) is not zero.
This type of function can represent complex relationships and is often used in calculus and other areas of mathematics. Understanding rational functions helps to analyze their behavior, such as finding asymptotes, intercepts, and points of discontinuity.

• When dealing with rational functions, a common goal is simplification, which often leads to partial fraction decomposition.
• Partial fraction decomposition breaks a rational function into a sum of simpler fractions.

Algebraic Fractions

Algebraic fractions are expressions that represent the division of one polynomial by another. They form the core part of rational functions and are crucial in algebra, calculus, and beyond.
Understanding and manipulating these fractions require skills in addition, subtraction, multiplication, and division, just like with ordinary fractions.

• To combine algebraic fractions, you must find a common denominator.
• Simplifying algebraic fractions often involves factoring polynomials and cancelling common factors.
• Sum, difference, product, and quotient rules apply, similar to numerical fractions, but with a focus on polynomial operations.

Polynomial Denominator Factors

In rational functions, the denominator polynomial can be factored into simpler terms, known as polynomial denominator factors. Recognizing these factors is essential for partial fraction decomposition.
Each distinct linear or irreducible quadratic factor in the denominator corresponds to a fraction term in the decomposition. For example, in the rational function \( \frac{1}{(x-1)(x+2)} \), the factors \((x-1)\) and \((x+2)\) each lead to a separate term in the decomposition.

• Each linear factor in the denominator yields a corresponding term with a constant numerator.
• For irreducible quadratic factors, the numerators in the decomposition are linear expressions.
• This factorization process is essential to transform a complex rational function into a more manageable form.