In Exercises 63–80, write the partial fraction decomposition...

In Exercises 63–80, write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$ \frac{x-2}{x^{2}+4 x+3} $$

Key Concepts

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials. Understanding rational expressions is key to simplifying, decomposing, and analyzing their behavior by looking at their domain, asymptotes, and intercepts.

Factoring Polynomials

Factoring polynomials is the process of rewriting a polynomial as a product of simpler polynomials. This fundamental skill is crucial for partial fraction decomposition, as it allows one to break down complex rational expressions by identifying and using the factors of the denominator.

Partial Fraction Decomposition

Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions whose denominators are the factors of the original denominator. This technique simplifies integration and helps understand the behavior of the function by breaking it into more manageable pieces.

Algebraic Verification

Algebraic verification is the process of checking that the sum of the decomposed fractions recombines to equal the original rational expression. This step ensures that the partial fraction decomposition is performed correctly, by carefully combining fractions and simplifying the result.

Graphical Verification

Graphical verification uses visual tools such as graphing utilities to compare the graph of the original rational expression with that of its decomposed partial fractions. If the graphs coincide, it offers additional confirmation that the decomposition is correct and the expressions are equivalent.

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