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 partial fraction decomposition of the given rational expression is \( \frac{A}{x^{2} + 7} + \frac{Bx+C}{(x^{2} + 7)^2} \)

Step 1: Identify the form of the partial fractions

First, identify the type of factors in the denominator. The denominator \( (x^{2}+7)^2 \) is a square of a binomial factor. This corresponds to the pattern of partial fraction \(\frac{A}{x^{2}+7} + \frac{Bx+C}{(x^{2}+7)^2} \). A, B and C are the constants.

Step 2: Write the partial fraction decomposition

Without finding the values of constants, write the partial fraction decomposition for the given expression. The partial fractions form of \( \frac{7x^{2} - 9x + 3}{(x^{2}+7)^2} \) is \( \frac{A}{x^{2} + 7} + \frac{Bx+C}{(x^{2} + 7)^2} \). This form is easier to integrate and simplify than the original function.

Key Concepts

Understanding Rational Expressions

A rational expression is essentially a fraction where the numerator and denominator are both polynomials. These expressions are vital in algebra because they extend the basic concept of fractions to expressions involving variables.
While a simple fraction might be \( \frac{3}{4} \), a rational expression can look like \( \frac{7x^2 - 9x + 3}{x^2 + 7} \).

• The numerator is any polynomial, such as \( 7x^2 - 9x + 3 \).
• The denominator is also a polynomial, such as \( (x^2 + 7)^2 \).
• Rational expressions can be analyzed, simplified, and decomposed using algebraic techniques.

Throughout algebra and calculus, working with rational expressions is essential for solving equations and understanding functions.

Exploring Denominator Factors

Understanding the denominator's factors is crucial for partial fraction decomposition. In our expression\( \frac{7x^2 - 9x + 3}{(x^2 + 7)^2} \), the denominator \( (x^2 + 7)^2 \) is particularly noteworthy.

• The expression \( x^2 + 7 \) is a binomial, and here, it is squared. Such a factor is termed as non-reducible quadratic because it doesn't factor down into linear factors over real numbers.
• For partial fraction decomposition, consider the original factor and its powers separately. In this case, we view both \( x^2 + 7 \) and \( (x^2 + 7)^2 \).
• When denominators involve squares, we account for each power in the decomposition. This approach helps in simplifying and working with the expressions.

This understanding allows us to correctly set up the form needed for decomposition.

Role of Constants in Fractions

In partial fraction decomposition, introducing constants allows us to break down complicated fractions into simpler parts. The goal is to express one large rational expression as a sum of simpler ones.
For the expression \( \frac{7x^2 - 9x + 3}{(x^2 + 7)^2} \), we use constants A, B, and C to set up the decomposition as:\[ \frac{A}{x^2 + 7} + \frac{Bx + C}{(x^2+7)^2} \]

• Each constant (A, B, C) represents a number or expression yet to be determined. They are crucial in adapting the decomposition to fit the original expression perfectly when recombined.
• These constants are placeholders we solve for in a full equation, allowing simplification and integration of the expression more easily.
• Identifying and working with these constants is a key step in both solving and understanding the behavior of rational expressions.

Partial fraction decomposition with these constants transforms complex polynomials and facilitates calculus operations such as integration.