Problem 8 - Algebra and Trigonometry

Question

\(3-12\) . 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

The partial fractions are \( \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1} \).

Step 1: Factor the Denominator

The first step is to factor the denominator. The expression given is \(x^4 - 1\), which can be recognized as a difference of squares. Thus: \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\). Further, \(x^2 - 1\) can be factored again as \((x - 1)(x + 1)\). Hence, the complete factorization is \((x - 1)(x + 1)(x^2 + 1)\).

Step 2: Set Up Partial Fraction Decomposition

With the factors of the denominator \((x - 1)(x + 1)(x^2 + 1)\), we can set up a partial fraction decomposition. The form is: \[ \frac{1}{(x-1)(x+1)(x^2+1)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx + D}{x^2+1} \]where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill that helps simplify mathematical expressions and solve equations. Consider the polynomial given in the exercise, \(x^4 - 1\). To factor this, we recognize it as a difference of squares, which is a special product of the form \(a^2 - b^2 =(a - b)(a + b)\). Here, \(x^4 - 1\) can be rewritten as \((x^2)^2 - 1^2\). By applying the difference of squares rule, we get:

• \(x^4 - 1 = (x^2 - 1)(x^2 + 1)\).

In the next step, \(x^2 - 1\) is further factored into \((x - 1)(x + 1)\) using the same difference of squares rule:

• Therefore, the complete factorization is \((x - 1)(x + 1)(x^2 + 1)\).

Understanding how to break down polynomials into their component factors is crucial, as it simplifies computations in calculus, algebra, and beyond.

Rational Expressions

Rational expressions involve ratios of polynomials, similar to simple numerical fractions but with polynomial expressions in the numerator and denominator. In the given exercise, we have a rational expression \(\frac{1}{x^4-1}\). The aim is often to simplify these expressions or set them up for partial fraction decomposition for integration and solving purposes.
To deal with such an expression, one must understand the implications of the factors in the denominator. Once we have factored \(x^4 - 1\) into \((x - 1)(x + 1)(x^2 + 1)\), each factor can potentially lead to a different form in the partial fraction decomposition.
Each linear factor, like \((x-1)\) and \((x+1)\), contributes a simple fraction term, whereas a quadratic factor, like \(x^2 + 1\), typically results in a fraction with a linear numerator, represented as \(Cx + D\).

• For instance, the decomposition takes the form: \(\frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx + D}{x^2+1}\).

Rational expressions require careful handling of factors to avoid undefined operations or incorrect simplifications.

Difference of Squares

The difference of squares is a powerful algebraic tool for factoring expressions that appear in the form \(a^2 - b^2\). This pattern simplifies into two binomials: \((a - b)(a + b)\).
In the context of the exercise, the expression \(x^4 - 1\) fits this pattern where \(a = x^2\) and \(b = 1\). Identifying the difference of squares allows us to quickly factor the polynomial into \((x^2 - 1)(x^2 + 1)\).

• \(x^2 - 1\) can also be further factored, as it's another instance of a difference of squares: \((x - 1)(x + 1)\).

Learning to spot the difference of squares is useful because it helps simplify expressions easily and reduces the complexity involved in solving equations. Being familiar with this technique is invaluable not only for algebra but also for calculus, where solving more complex expressions is often necessary.