Problem 8 - College Algebra

Question

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{7 x^{2}-9 x+3}{\left(x^{2}+7\right)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational function is \[\frac{A}{x^2 + 7} + \frac{Bx + C}{(x^2 + 7)^2}\] where A, B and C are constants that would be determined by equating coefficients and possibly solving systems of equations.

Step 1: Factor the denominator

The given rational function's denominator is \((x^2 + 7)^2\). It is already factored, with a repeated factor of \(x^2 + 7\).

Step 2: Set up the generic form of the decomposition

Each factor in the denominator will correspond to a term in the partial fraction decomposition. Since \(x^2 + 7\) is repeated twice in the denominator, there will be two terms in the decomposition. The form is \[\frac{A}{x^2 + 7} + \frac{Bx + C}{(x^2 + 7)^2}\] where A, B and C are constants to be determined.

Step 3: Understanding the result

This is the general form of the partial fraction decomposition for the given rational function. If we were to solve for A, B and C, then the original function could be rewritten as a sum of these much simpler fractions.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator consist of polynomials. In mathematical terms, a rational expression is typically represented as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).

Just like with numerical fractions, it is often useful to simplify or break down these expressions into simpler parts. This is particularly beneficial when you're trying to integrate, differentiate, or solve equations involving these expressions.

• A rational expression's complexity is in how the numerator and denominator polynomials interact.
• Managing these interactions often involves techniques like factoring and partial fraction decomposition.

By breaking down rational expressions into simpler components, calculations become more manageable, making it easier to understand their behavior and solve related problems.

Factoring

Factoring is an essential preliminary step when working with rational expressions and specifically when applying partial fraction decomposition. This involves breaking down a polynomial into its simplest form, expressed as a product of its factors. In this exercise, the denominator \((x^2 + 7)^2\) is already factored.

A polynomial in standard form is expressed as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Factoring it involves finding the expression of polynomials such that\

• They multiply to yield the original polynomial.
• They are as simple as possible.

Factoring allows us to see the structure of a polynomial expression. For instance, a squared term in the denominator, \((x^2 + 7)^2\), indicates that the expression can become a candidate for partial fraction decomposition later.

Repeated Factors

Repeated factors occur in situations where a particular factor in the denominator is raised to a power greater than one. In the given exercise, the factor \(x^2 + 7\) repeats, resulting in \((x^2 + 7)^2\).

Understanding repeated factors is key to setting up a correct partial fraction decomposition. Each occurrence of a repeated factor contributes additional terms to the decomposition, ensuring every possible simpler form is accounted for. In practice, for a repeated factor \(a\) that appears \(n\) times in the denominator:

• You must create a separate fraction for each power of \(a\) from 1 through \(n\).
• Each term in the decomposition is of the form: \( \frac{B}{a^i} \) where \(i\) is from 1 to \(n\).

In our exercise, this results in terms involving \(\frac{A}{x^2 + 7} \) and \( \frac{Bx + C}{(x^2 + 7)^2} \), covering both powers of the factor \(x^2 + 7\).

Constants Determination

Once the form of a partial fraction decomposition is established, constants must be assigned to each term. These constants ensure the decomposition accurately reconstructs the original rational expression if summed back together. In our exercise, these are represented as \(A\), \(B\), and \(C\).

Determining these constants usually involves:

• Multiplying through by the entire original denominator to eliminate the fractions.
• Equating coefficients from both sides of the equation for any matching powers of \(x\).
• Solving this resulting system of equations to find the values of the constants.

This process simplifies complex rational functions into sections that are easier to integrate, differentiate, or evaluate individually. While the task doesn't require solving for \(A, B,\) and \(C\), understanding that this process exists allows one to appreciate the utility of partial fraction decomposition in simplifying calculations.