Problem 64 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. Use an extension of the Great Question! on page 859 to describe how to set up the partial fraction decomposition of a rational expression that contains powers of a prime cubic factor in the denominator. Give an example of such a decomposition.

Step-by-Step Solution

Answer

This task makes sense. Understanding the partial fraction decomposition of a rational expression that contains powers of a prime cubic factor in the denominator is essential for simplifying algebraic expressions and solving differential equations. The decomposition is expressed as a sum of simpler fractions. A clear example is \( R(x) = A/x + B/x^2 + C/x^3 \) for \( R(x) = (x^2 + 3x + 2)/(x^3 + 3x^2 + 3x + 1) \).

Step 1: Understanding Key Terms

A rational expression is a fraction where both the numerator and the denominator are polynomials. A prime cubic factor is an irreducible polynomial of degree 3 (cubic polynomial). Now to do a partial fraction decomposition means to express a complicated fraction as a sum of simpler fractions (partial fractions).

Step 2: Setting Up the Decomposition

To set up the partial fraction decomposition with a prime cubic factor in the denominator, suppose the main rational expression is \( R(x) = \frac{P(x)}{Q(x)} \), where \( Q(x) \) is a product of distinct prime cubic factors. Let \( Q(x) = (a_1 + b_1)(a_2 + b_2)^2(a_3 + b_3)^3 \ldots \). Then, the partial fraction decomposition of \( R(x) \) can be written as follows: \( R(x) = \frac{A}{a_1 + b_1} + \frac{B}{(a_2 + b_2)^2} + \frac{C}{(a_2 + b_2)} + \frac{D}{(a_3 + b_3)^3} + \frac{E}{(a_3 + b_3)^2} + \frac{F}{a_3 + b_3} + \ldots \) where A, B, C, D, E, F are constants to be determined.

Step 3: Providing an Example

Suppose \( R(x) = \frac{x^2 + 3x + 2}{x^3 + 3x^2 + 3x + 1} \). Here, the denominator is a cubic with no reducible part. The partial fraction decomposition can be written as \( R(x) = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} \). Then, coefficients A, B, and C can be found by equating the numerators, yielding a system of linear equations, and solving it.

Key Concepts

Rational Expressions

A rational expression, in the realm of algebra, is akin to a fraction, but instead of common numbers, its components are polynomials. Just like the fractions we use in basic arithmetic, a rational expression is represented as one polynomial divided by another, denoted as \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

Understanding rational expressions is a crucial skill in algebra, as they often require manipulation to simplify, analyze, or solve equations. When it comes to partial fraction decomposition - a method used to break down complex rational expressions into simpler parts - grasping the definition of a rational expression becomes even more vital.

Imagine trying to tackle a math problem that looks overwhelming due to the size of the fraction involved; that's where partial fraction decomposition steps in. It enables students to convert this intimidating expression into a series of easier, more manageable pieces. It's a bit like taking a large, complex puzzle and dividing it into smaller sections that are simpler to solve individually.

Prime Cubic Factor

When delving into partial fraction decomposition involving more advanced polynomials, students may encounter what is called a 'prime cubic factor.' In essence, this factor is a cubic polynomial that cannot be broken down into simpler polynomial factors—in other words, it's irreducible.

A cubic polynomial is one where the highest power of the variable, typically \( x \), is three. A prime cubic factor, however, stands alone without the ability to be factored further. For example, the polynomial \( x^3 + 3x^2 + 3x + 1 \) is a prime cubic factor because it does not have any roots which can be factored out.

In the context of partial fractions, recognizing prime cubic factors is key. If the denominator of a rational expression includes irreducible cubic terms, the decomposition will incorporate these terms unchanged. Each term needs its own unique fractional component within the decomposition. This specificity ensures that each piece of the fraction can be carefully adjusted to form the original complex expression when recombined.

Polynomials

Polynomials are the bread and butter of algebra, a fundamental concept that students encounter in various forms. They consist of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.

Represented generally as \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0 \) where \( a_n\), \( a_{n-1}\), ..., \( a_0 \) are constants, and \( n \) is a non-negative integer, a polynomial can range from being very simple, such as \( x \) or \( x^2 + 4 \) to incredibly complex. The highest power of \( x \) denotes the polynomial's degree.

When referring to the decomposition of rational expressions, polynomials play a crucial role since they form both the numerator and denominator of the expression. The process of partial fraction decomposition essentially involves breaking down these polynomial elements into components that are more straightforward to work with. Understanding the structure and behavior of polynomials is, therefore, an integral part of mastering partial fraction decomposition and simplifying complex rational expressions.