Problem 1 - Calculus Single Variable

Question

In each of Exercises \(1-10,\) write down the form of the partial fraction decomposition of the given rational function. Do not explicitly calculate the coefficients. \(\frac{2 x^{3}+x+1}{\left(x^{2}+1\right)\left(x^{2}+4\right)}\)

Step-by-Step Solution

Answer

\(\frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 4}\).

Step 1: Identify the Denominator

The given rational function is \(\frac{2x^3 + x + 1}{(x^2 + 1)(x^2 + 4)}\). The denominator \((x^2 + 1)(x^2 + 4)\) is already factored into irreducible quadratic factors over the real numbers.

Step 2: Set Up the Partial Fraction Decomposition

For each quadratic factor \((x^2 + a^2)\), we assign a linear numerator \(ax + b\). Thus, the partial fraction decomposition for \((x^2 + 1)(x^2 + 4)\) will be in the form: \(\frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 4}\).

Step 3: Formulate the Final Expression

Compile the expression for the partial fraction decomposition: \[\frac{2x^3 + x + 1}{(x^2 + 1)(x^2 + 4)} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 4}.\] This setup shows the form of the partial fraction decomposition without solving for the coefficients \(A, B, C,\) and \(D\).

Key Concepts

Rational Functions

Rational functions are expressions that represent the quotient of two polynomials. They take the general form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are significant because they appear frequently in calculus and algebra problems.

• The numerator \( P(x) \) can be any polynomial.
• The denominator \( Q(x) \) must not be zero, because division by zero is undefined.
• Rational functions can have vertical asymptotes where the denominator equals zero, which indicate points of discontinuity.

In the exercise, the given rational function \( \frac{2x^3 + x + 1}{(x^2 + 1)(x^2 + 4)} \) demonstrates a polynomial numerator of degree three and a more complex polynomial denominator consisting of irreducible quadratic factors. Understanding how to break down these rational functions into simpler components, like partial fractions, is crucial for integration and solving real-life application problems.

Irreducible Quadratic Factors

Irreducible quadratic factors are quadratic expressions that cannot be factored over the real numbers into linear terms. They often appear in the factorization of polynomial denominators of rational functions.

• These factors are of the form \( x^2 + a^2 \) where \( a \) is a real number and the expression does not have real roots.
• An example is \( x^2 + 1 \), which can only be factored into linear terms with complex numbers (\( (x + i)(x - i) \)).

The exercise includes the rational function \( \frac{2x^3 + x + 1}{(x^2 + 1)(x^2 + 4)} \), where both quadratic factors are irreducible over the real numbers. These factors play a key role in the partial fraction decomposition process, as each irreducible quadratic factor corresponds to a term with a linear numerator in the decomposition.

Linear Numerator

In partial fraction decomposition, a linear numerator refers to the expression represented with variables having a degree of at most one. When dealing with irreducible quadratic factors in the denominator, each term in the decomposition includes a linear numerator.

• A linear numerator is expressed as \( ax + b \), where \( a \) and \( b \) are constants.
• This setup allows the decomposed terms to be in the simplest form, making it easier to solve for the coefficients later, if needed.

In solving the exercise \( \frac{2x^3 + x + 1}{(x^2 + 1)(x^2 + 4)} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 4} \), \( Ax + B \) and \( Cx + D \) are the linear numerators assigned to the irreducible quadratic factors in the denominator. This decomposition format does not require solving for specific values of \( A, B, C, \) and \( D \) but provides a structured way to break the function into simpler parts.