Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Write out the partial fraction decomposition of each rational function. You need not determine the coefficients; just set them up. (a) \(\frac{x^{2}+3}{x(x-1)(x+5)}\) (b) \(\frac{x}{x^{3}+x}\)

Step-by-Step Solution

Answer

The required decomposition of rational functions are as follows: (a) \( \frac{x^{2}+3}{x(x-1)(x+5)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+5} \) and (b) \( \frac{x}{x^{3}+x} = \frac{A}{x} + \frac{Bx + C}{x^{2} + 1} \) where A, B, C are constants to be determined.

Step 1: Decomposition of Rational Function (a)

Given the rational function \( \frac{x^{2}+3}{x(x-1)(x+5)} \), its denominator has 3 factors: x, (x-1), (x+5), all of which are distinct linear. Thus, the rational function can be decomposed into the form \[ \frac{x^{2}+3}{x(x-1)(x+5)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+5} \] where A, B, C are constants to be determined.

Step 2: Decomposition of Rational Function (b)

Given the rational function \( \frac{x}{x^{3}+x} \), its denominator can be factored as \( x^{3}+x = x(x^{2}+1) = x(x+i)(x-i)\), where 'i' is square root of -1 and x, (x+i), (x-i) are its factors. Since, in real number system, factors (x+i) and (x-i) can be treated as a single irreducible quadratic \( x^{2}+1 \), the rational function can be decomposed into the form \[ \frac{x}{x^{3}+x} = \frac{A}{x} + \frac{Bx + C}{x^{2} + 1} \] where A, B, C are constants to be determined.

Key Concepts

Rational Functions

Rational functions are quotients of two polynomials. They can be written in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The polynomial in the numerator may have a degree that is less than or equal to the degree of the polynomial in the denominator. Rational functions are important because they appear frequently in calculus and modeling scenarios.
To understand rational functions better, consider how the denominator influences the overall function. If the denominator has a degree greater than the numerator, the function generally decreases as \( x \) moves towards infinity.
When simplifying or working with rational functions, particularly when decomposing them into partial fractions, we often focus on factoring the denominator first.

Linear Factors

Factors of a polynomial that take the form \((ax + b)\) are called linear factors. Linear factors are crucial in partial fraction decomposition because they represent simple, distinct parts of the polynomial that can be broken down individually.
In the decomposition process, each linear factor in the denominator corresponds to a separate fraction in the decomposition of the rational function. For example, in exercise (a):

• The rational function \( \frac{x^{2}+3}{x(x-1)(x+5)} \) has a denominator composed of linear factors \( x \), \( (x-1) \), and \( (x+5) \).
• The partial fraction setup involves writing separate terms for each factor: \( \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+5} \).

In these partial fractions, each constant \( A, B, C \) will be determined to match the original function.

Irreducible Quadratic

An irreducible quadratic is a term that cannot be factored further into linear factors using real numbers. In partial fraction decomposition, these appear as a single entity instead of separate linear factors.
An example is present in exercise (b), where the denominator \( x^3 + x \) can be factored in the complex number system, but when dealing with real numbers, it's represented as \( x(x^2 + 1) \).
Since \( x^2 + 1 \) has no real roots, it is considered an irreducible quadratic. As such, the partial fraction decomposition involves a term of the form \( \frac{Bx + C}{x^2 + 1} \) to account for both terms originally present in the numerator and to maintain dimensional integrity in the equation.

Constants Determination

Once the partial fraction has been set up, the next step is to determine the unknown constants. This process involves ensuring the decomposed form is equivalent to the original rational function.

• Set up an equation for each part of the numerator after equating the decomposed version and original rational function.
• Substitute suitable values for \( x \) to simplify the equations, helping isolate and solve for each constant.
• Equating coefficients of similar terms is another method, allowing you to find values for the constants \( A, B, C \) in the setup from exercise (a) and \( b \).

Applying these techniques systematically lets you solve for the unknowns, ensuring that the transformed function, when combined, represents the original rational function accurately.