Problem 1 - Algebra and Trigonometry

Question

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \( \frac{A}{x - 2} + \frac{B}{x + 1} \).

Step 1: Define the Form of the Partial Fraction Decomposition

Given a rational expression \( \frac{11x - 10}{(x - 2)(x + 1)} \) , we can write its partial fraction decomposition as \( \frac{A}{x - 2} + \frac{B}{x + 1} \) because \( (x - 2) \) and \( (x + 1) \) are distinct linear factors.

Step 2: Identification of Constants

We do not need to solve for the constants A and B. Thus the partial fraction decomposition in terms of its general form is \( \frac{A}{x - 2} + \frac{B}{x + 1} \).

Key Concepts

Rational Expressions

Rational expressions are fractions that have polynomials in both their numerator and denominator. They are just like numeric fractions but with variables included.

Rational expressions can be complex and aren't always in the simplest form for solving equations or evaluating. That's where partial fraction decomposition comes into play. This method breaks down complex rational expressions into simpler, more manageable pieces, which can often make integration, differentiation, and other algebraic procedures much easier.

An example of a rational expression is the equation given in the exercise: \[ \frac{11x - 10}{(x - 2)(x + 1)} \]. Here, the numerator is a linear polynomial (11x - 10) and the denominator consists of two linear factors, (x - 2) and (x + 1), multiplied together.

Linear Factors

In algebra, linear factors are expressions of the first degree, which means they have no exponents higher than one. These factors are the building blocks of polynomials and are generally in the form of (ax + b), where 'a' and 'b' are constants and 'x' is the variable.

The rational expression from the exercise is composed of such linear factors in the denominator. Each factor, (x - 2) and (x + 1), represents a potential 'zero' of the polynomial where the expression will be undefined, hence the importance of recognizing these components.

In partial fraction decomposition, we focus on these linear factors to rewrite the complex fraction as a sum of simpler fractions, each having one of these factors in the denominator.

Algebraic Techniques

To tackle problems in algebra, especially when dealing with rational expressions, a variety of algebraic techniques can be employed. Partial fraction decomposition is one such technique that is particularly useful when integrating rational functions.

The process involves breaking down a fraction into a sum of fractions with simpler denominators. For the given example, the initial step is to express the decomposition form \[ \frac{11x - 10}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \], where 'A' and 'B' are constants to be determined. This setup is essential for solving the integral of rational expressions that do not initially appear easy to integrate.

To fully perform partial fraction decomposition, you would equate numerators and solve for 'A' and 'B', although this step is beyond the current exercise scope. Recognizing which algebraic techniques to apply in various contexts is a core skill in advancing through algebra.