Problem 2 - Calculus Single Variable

Question

Write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{4}}{\left(x^{2}+2 x+2\right)\left(2 x^{2}+5 x+3\right)}\)

Step-by-Step Solution

Answer

The decomposition is \(\frac{Ax + B}{x^2 + 2x + 2} + \frac{Cx + D}{2x^2 + 5x + 3}\).

Step 1: Recognize the Denominator

First, observe the denominator of the rational function \((x^2 + 2x + 2)(2x^2 + 5x + 3)\). Notice that it is a product of two distinct quadratic polynomials.

Step 2: Set Up Partial Fractions Structure

Since both denominators \((x^2 + 2x + 2)\) and \((2x^2 + 5x + 3)\) are irreducible quadratics, the partial fraction decomposition will have the form: \[ \frac{2x^4}{(x^2 + 2x + 2)(2x^2 + 5x + 3)} = \frac{Ax + B}{x^2 + 2x + 2} + \frac{Cx + D}{2x^2 + 5x + 3} \].

Key Concepts

Rational Function

A rational function is essentially a fraction where both the numerator and the denominator are polynomials. Rational functions take the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not equal to zero. These functions can describe a wide variety of behaviors depending on the degrees and coefficients of the polynomials involved. One interesting aspect of rational functions is how they graph. Depending on the numerator and denominator, the graph can have vertical asymptotes, horizontal asymptotes, or even oblique asymptotes.

• Vertical asymptotes occur where the denominator equals zero and the numerator is not also zero.
• Horizontal asymptotes are determined by the degrees of the numerator and denominator polynomials.
• Oblique asymptotes may occur when the degree of the numerator is exactly one more than the degree of the denominator.

Ultimately, understanding the structure and properties of rational functions is essential for many areas such as calculus and engineering.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, and it is usually expressed in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratics play a fundamental role in algebra and appear frequently in mathematical problems.Key characteristics of quadratic polynomials include:

• Standard Form: Can be written as \( ax^2 + bx + c \).
• Factoring: Quadratics can sometimes be factored into two binomials (e.g., \((x + p)(x + q)\)). Factoring is possible when the roots of the equation are real and rational.
• The Discriminant: The expression \( b^2 - 4ac \) determines the nature of the roots. If it's positive, there are two distinct real roots; if zero, one real root (a repeated root); and if negative, the roots are complex and conjugate.
• Graph: Quadratics graph as a parabola, opening upward if \( a > 0 \) and downward if \( a < 0 \).

Quadratic polynomials are central in calculus for optimization and modeling problems, making them essential to master.

Irreducible Quadratics

Irreducible quadratics are quadratic polynomials that cannot be factored into real polynomials of lower degree. In other words, these are polynomials for which the discriminant \( b^2 - 4ac \) is negative, meaning they have no real roots.Key features include:

• Since the roots are complex, irreducible quadratics cannot be broken down into real, simpler polynomials.
• They often appear in the denominator when performing partial fraction decomposition.
• In a partial fraction decomposition, each irreducible quadratic factor in the denominator is associated with a linear term in the numerator.
• For a quadratic \( ax^2 + bx + c \) where the discriminant is negative, its roots are of the form \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), indicating complex conjugates.

Understanding irreducible quadratics is crucial when dealing with real analysis problems, such as integration using partial fractions.

Denominator Decomposition

Denominator decomposition is a pivotal technique used in simplifying rational expressions, especially when using partial fraction decomposition. It involves expressing the denominator of a rational function as a product of simpler factors, where possible. Steps to perform denominator decomposition include:

• Identify Factors: Break down the denominator into its irreducible factors. For instance, in quadratic cases, this means determining if it can be factored into real roots or if it remains irreducible.
• Consider Repeated Factors: If any factors repeat, you'll need to account for them by including terms in the decomposition for each power.
• Apply to Partial Fractions: Use the factors found in the decomposition to set up a structure for the partial fractions. Each factor or group of factors becomes a separate term in the decomposition.

An understanding of denominator decomposition not only simplifies complex rational functions but also forms the basis for evaluating integrals, solving differential equations, and other advanced mathematics problems.