Problem 6 - 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. $$\frac{1}{x^{4}-x^{3}}$$

Step-by-Step Solution

Answer

The expression is \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}\).

Step 1: Factor the Denominator

First, factor the denominator \(x^4 - x^3\). Notice that we can factor out an \(x^3\) term, giving us \(x^3(x-1)\).

Step 2: Identify the Partial Fractions

Based on the factored form \(x^3(x-1)\), identify the structure of the partial fraction decomposition. We will have terms corresponding to each factor and repeated factors will have additional terms:- \(\frac{A}{x}\)- \(\frac{B}{x^2}\)- \(\frac{C}{x^3}\)- \(\frac{D}{x-1}\)

Step 3: Write the Partial Fraction Decomposition

Combine the terms from Step 2 into the full partial fraction decomposition. The expression becomes:\[\frac{1}{x^4 - x^3} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a number into its prime factors, but with polynomial expressions. It allows us to express a polynomial as a product of its simplest parts. In our example, the polynomial is the denominator of the function, \( x^4 - x^3 \). To factor it, we look for common factors first. Here, both terms \( x^4 \) and \( x^3 \) share an \( x^3 \), which can be factored out:

\[ x^4 - x^3 = x^3(x - 1) \]

This means the expression is now a product of \( x^3 \) and \( (x - 1) \). Factoring makes the expression simpler and easier to work with, especially for partial fraction decomposition, where we need to identify individual factors. Always check your factoring by expanding it back to ensure it matches the original polynomial.

Linear Factors

Linear factors are components of a polynomial that are of the first degree, meaning their highest exponent of a variable is 1. In our example, when \( x^3(x - 1) \) is factored, \( (x - 1) \) is a linear factor because it is of the form \( ax + b \) where \( a \) and \( b \) are constants, and the highest power of \( x \) is 1.

Linear factors are important in the partial fraction decomposition as they indicate terms that will appear in the decomposition. For each linear factor, there will be a corresponding term in the decomposition. In this case, \( \frac{D}{x-1} \) corresponds to the linear factor \( (x - 1) \). Clearly identifying these parts is crucial to setting up and solving the decomposition correctly.

Repeated Factors

Repeated factors occur when a polynomial has a term that appears more than once, raised to a power greater than 1. In the expression \( x^3(x - 1) \), \( x \) is a repeated factor since it is raised to the third power. This affects how we set up our partial fraction decomposition.

For repeated factors, each exponent will have its own term in the decomposition. This means if we have \( x^n \), we will have partial fractions like:

• \( \frac{A}{x} \)
• \( \frac{B}{x^2} \)
• \( \frac{C}{x^3} \)

This principle applies to our example, where \( x^3 \) gives rise to three separate terms, each with its own constant in the decomposition. Recognizing repeated factors allows us to assign the correct number of terms in the decomposition and ensures that each part gets appropriate attention.