Problem 8 - 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. $$\frac{x^{4}+x^{2}+1}{x^{2}\left(x^{2}+4\right)^{2}}$$

Step-by-Step Solution

Answer

The decomposition is: \( \frac{A}{x} + \frac{B}{x^2} + \frac{Cx+D}{x^2+4} + \frac{Ex+F}{(x^2+4)^2} \).

Step 1: Identify the Denominator Components

The denominator of the given function is \( x^2 (x^2 + 4)^2 \). It comprises two distinct parts: \( x^2 \), which contributes two linear factors, and \((x^2 + 4)^2\), which contributes two irreducible quadratic factors repeated twice.

Step 2: Set Up the Partial Fraction Decomposition Structure

According to partial fraction decomposition rules, each term in the denominator needs its own distinct numerator form. For the \( x^2 \) term, we assign fractions of the form \( \frac{A}{x} + \frac{B}{x^2} \). For the \( (x^2 + 4)^2 \) term, where the quadratic term is repeated, we use numerators of the linear form, giving \( \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2} \).

Step 3: Write the Full Decomposition Form

Combine all the fractions from Step 2 into a single expression: \[ \frac{x^4 + x^2 + 1}{x^2(x^2 + 4)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{x^2 + 4} + \frac{Ex + F}{(x^2 + 4)^2} \].This represents the partial fraction decomposition structure without solving for the coefficients.

Key Concepts

Polynomial Functions

Polynomial functions form the backbone of many algebraic expressions. They consist of variables raised to non-negative integer exponents and coefficients. A polynomial function is typically written in the form:

• Coefficients: These are the numbers in front of the variable terms, e.g., 4 in 4x².
• Terms: These are the parts of the polynomial separated by addition or subtraction, e.g., 4x² + 3.
• Degree: Determined by the highest exponent of the variable, indicating the function's behavior as the independent variable changes.

Understanding these elements is crucial because they dictate the complexity of a polynomial function's shape on a graph.
For instance, the degree can tell us the number of times the graph can cross the x-axis or how many turning points it could have.

Rational Expressions

Rational expressions combine the aspects of polynomials within fractions. They have a numerator and a denominator, both of which are polynomial functions. These expressions are called 'rational' because they form a ratio.

• Expressions like \(\frac{x+1}{x-2}\) or \(\frac{x^2+3}{x^3-x}\) are common examples.
• They must be simplified carefully, ensuring that the denominator never equals zero to avoid undefined expressions.

Rational expressions create a rich area of study in algebra since manipulating them often involves complex operations such as factoring or finding a least common denominator. Understanding them is key to dealing with many algebraic equations and functions.

Algebraic Fractions

Algebraic fractions are another term for expressions that involve polynomials within a fraction structure. They can be simplified, expanded, or decomposed through various operations. This often involves setting the polynomial equal to zero to find roots:

• Simplifying: Finding the greatest common factor (GCF) to reduce fractions to their simplest forms.
• Expanding: Multiplying factors to write as a single polynomial.
• Decomposing: Breaking down complex fractions into simpler parts, such as the task in partial fraction decomposition.

These operations form the foundation of manipulating algebraic equations, making algebraic fractions an essential topic not only in mathematics but in fields such as engineering and physics as well.

Irreducible Quadratic Factors

Irreducible quadratic factors are quadratic expressions that cannot be factored further using real numbers. A typical quadratic is \(ax^2 + bx + c\), but irreducible forms do not yield 'nice' numbers when factored:

• For example, \(x^2 + 4\) cannot be factored into real-number solutions.
• These factors often accompany complex numbers when solving for roots, involving imaginary numbers.

They hold importance in partial fraction decomposition when breaking down rational expressions into simpler components. Recognizing and handling irreducible quadratic factors effectively transforms complex algebraic work into more actionable forms, which is necessary for deeper mathematical analysis and understanding.