Problem 6 - Precalculus: Mathematical 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. $$\frac{1}{x^{4}-1}$$

Step-by-Step Solution

Answer

Decompose using \( \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1} \).

Step 1: Factor the Denominator

First, we need to factor the expression in the denominator: \( x^4 - 1 \). This is a difference of squares, \( x^4 - 1 = (x^2 - 1)(x^2 + 1) \). We further factor \( x^2 - 1 \) as \( (x-1)(x+1) \). Thus the factorization is \( (x-1)(x+1)(x^2+1) \).

Step 2: Set Up the Partial Fractions

Based on the factorization, we can decompose the function into partial fractions. For the linear factors \((x-1)\) and \((x+1)\), we assign constant numerators: \( \frac{A}{x-1} + \frac{B}{x+1} \). For the irreducible quadratic factor \((x^2+1)\), we assign a linear numerator: \( \frac{Cx+D}{x^2+1} \).

Step 3: Combine Partial Fractions into a Single Fraction

The form of the partial fraction decomposition is: \[ \frac{1}{x^4-1} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1}. \] Here, \( A \), \( B \), \( C \), and \( D \) are constants that you would solve for normally by combining these fractions into a single fraction equal to the original and equating numerators.

Key Concepts

Polynomial Factorization

Polynomial Factorization is a process crucial for simplifying expressions and solving equations. In our exercise, we have a polynomial in the denominator. Specifically, it is the term \( x^4 - 1 \). The first step in simplifying a fraction like this is to factor the polynomial.

This particular polynomial belongs to a category called the 'difference of squares', thanks to its similarity to the general form \( a^2 - b^2 \). The equation can be factored as \[ x^4 - 1 = (x^2 - 1)(x^2 + 1). \]Next, we notice that \( x^2 - 1 \) itself is another difference of squares, expressible as \[ (x-1)(x+1). \]
Putting it all together, the complete factorization of \( x^4 - 1 \) becomes\[ (x-1)(x+1)(x^2+1).\]
Factoring polynomials allows us to break them into simpler parts, making calculations or further algebraic manipulations more manageable. This concept is foundational in the study of algebra.

Rational Functions

Rational Functions are expressions that represent the ratio of two polynomials. In our context, the function \( \frac{1}{x^4 - 1} \) is a rational function where the numerator is 1, and the denominator is the polynomial \( x^4 - 1 \).

Understanding rational functions is essential, especially since these functions can often be decomposed into simpler parts. Such decomposition offers a clearer insight into the behavior of the function near specific values or points. The act of breaking down these functions into simpler 'partial fractions' provides a powerful tool for integration and other operations in calculus.

Overall, rational functions, like the one in our exercise, open up opportunities for deeper analysis across various branches of mathematics, including algebra, calculus, and beyond.

Algebraic Fractions

Algebraic Fractions involve expressions with polynomials in the numerator or denominator or both, similar to normal fractions but with polynomial expressions. In the given exercise, \( \frac{1}{x^4 - 1} \) is an algebraic fraction that can be decomposed into partial fractions.

When dealing with algebraic fractions, especially complex ones, we aim to express them as the sum of simpler fractions that are easier to work with. For decomposition, like in our exercise, once the denominator is factored, we assign a partial fraction to each factor. Linear factors get constant numerators, such as \( \frac{A}{x-1} \) or \( \frac{B}{x+1} \). For irreducible quadratic factors, a more complex numerator like \( \frac{Cx+D}{x^2+1} \) is used.

Decomposing algebraic fractions into their partial fractions is a strategic move to simplify integration, solving differential equations, and other calculus operations. It transforms a complex problem into a series of simpler ones, each manageable in their way.