Problem 36 - College Algebra Essentials

Question

write the partial fraction decomposition of each rational expression. $$ \frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8} $$

Step-by-Step Solution

Answer

\(\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8} = \frac{-1}{x+1} + \frac{3x+5}{x^2+x+8}\)

Step 1: Factor the Denominator

First, factor the denominator \(x^{3}+2 x^{2}+4 x+8 = (x+1)(x^2+x+8)\). The quadratic factor here is irreducible.

Step 2: Set Up the Partial Fractions

Express the rational expression in terms of its partial fractions as: \(\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8} = \frac{A}{x+1} + \frac{Bx+C}{x^2+x+8}\)

Step 3: Determine the Values of A, B and C

To find the values of A, B, and C, multiply both sides by the denominator on the right to eliminate the denominators, then equate coefficients on both sides.

Step 4: Solve the Resulting System of Equations

Solving the system of equations gives the values A=-1, B=3 and C=5.

Step 5: Write Down the Final Answer

Substitute the values of A, B, and C into the equation from step 2. This gives \(\frac{3 x^{2}-2 x+8}{x^{3}+2 x^{2}+4 x+8} = \frac{-1}{x+1} + \frac{3x+5}{x^2+x+8}\)

Key Concepts

Rational Expressions

Rational expressions are similar to fractions but involve polynomials in their numerators and/or denominators. They can represent a variety of algebraic forms and are fundamental to understanding many of the more advanced concepts in algebra and calculus.

In rational expressions, it's crucial to ensure that the denominator is not zero. When simplifying these expressions, our goal is often to break them down into simpler parts, much like simplifying normal fractions.

• The numerator and the denominator are polynomials.
• If the degree of the numerator is equal to or higher than that of the denominator, a long division is usually performed first.
• Rational expressions can be added, subtracted, multiplied, and divided (except division by zero).
• Key operations also include factoring to simplify or break down the expressions further.

Understanding rational expressions helps in converting complex expressions into simpler terms, as well as in calculating limits and analyzing graphs of functions.

Factoring Denominators

Factoring denominators is one of the initial and crucial steps in the process of working with rational expressions, especially when we need to decompose them into partial fractions.

The core idea is to break down the polynomial in the denominator into a product of simpler polynomials or linear factors.

• Identify and extract the greatest common factor, if there is one, for the entire expression.
• Check for common patterns, like the difference of squares, perfect squares, or cubes, which can be factored directly.
• For higher-degree polynomials, other methods like synthetic division, polynomial division, or even the quadratic formula might be necessary.

Factoring makes it much easier to then set up the expression for partial fraction decomposition, as each factor represents a potential partial fraction term. In the example provided, the denominator is factored into \((x+1)(x^2+x+8)\), where the quadratic factor is irreducible over the real numbers.

Systems of Equations

Systems of equations involve multiple equations that relate to each other, and we often solve them to find specific values of unknown variables. In the context of partial fraction decomposition, once we set up the partial fractions, we simplify this into a system of linear equations.

This process involves:

• Expressing the original rational expression in terms of its partial fractions.
• Multiplying through by the common denominator to eliminate fractions.
• Equating the coefficients of the polynomials on both sides of the equation.
• Solving the resulting system for the variables, often using substitution or elimination methods.

In the given solution, the system derived from the decomposition, after clearing fractions and equating coefficients, gives us precise values for \(A\), \(B\), and \(C\). This system helps us determine how each part of our fraction relates to others and builds towards the solution of simplifying a complex rational expression.

Algebraic Fractions

Algebraic fractions are essentially expressions with numerators and denominators that are algebraic expressions, usually polynomials. Understanding them involves understanding traditional operations on numbers, but within the framework of algebraic expressions.

Key points to remember include:

• Simplifying expressions using factoring techniques or polynomial division.
• Knowing how to handle additions, subtractions, multiplications, and divisions with these fractions.
• Being aware of restrictions in the domain, such as values that make the denominator zero, as they make the expression undefined.
• Using algebraic manipulation to transform these fractions into forms that are easier to work with, such as converting them into partial fractions.

Algebraic fractions are a fundamental aspect of algebra, helping us to understand how complex expressions can be manipulated and simplified, as demonstrated in the partial fraction decomposition example where the seemingly complex fraction is expressed in a more manageable form. By converting them into partial fractions, we find simplified versions which make subsequent mathematical operations easier.