Problem 8 - College Algebra Essentials

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 expression is: \(\frac{A}{x^{2}+7} +\frac{Bx+C}{(x^{2}+7)^{2}}\).

Step 1: Identify the Factors

The denominator in the given fraction expression is \((x^{2}+7)^{2}\), which has repeated quadratic factors. Note that \((x^{2}+7)\) cannot be broken down further as it does not have real roots.

Step 2: Write Down the Form of Partial Fraction Decomposition

According to the aforementioned analysis, for any repeated factors, we need to include terms up to the highest power of those factors in the decomposition. Considering that our repeated factor here is of degree 2, the form of the partial fraction decomposition would look like:\[ \frac{A}{x^{2}+7} +\frac{Bx+C}{(x^{2}+7)^{2}}\]This represents our final decomposed form of the given rational function. \(A\), \(B\), and \(C\) are constants which could be solved if it were necessary.

Key Concepts

Rational Expressions

Rational expressions are fractions that have polynomials in both their numerator and denominator. Much like the fractions you encounter with numbers, rational expressions can be simplified, added, subtracted, multiplied, and divided. However, they can also be more complex, which is why we often turn to techniques such as partial fraction decomposition.

This strategy is particularly helpful when integrating rational expressions or finding their limits. Understanding rational expressions is crucial in calculus and algebra, as they frequently appear in various mathematical contexts.

Factorization

Factorization is a process of breaking down a complex expression into a product of simpler factors. In the context of rational expressions, factorization usually refers to the denominator, since it is essential for partial fraction decomposition.

To factor a polynomial, one must look for common factors, differences of squares, perfect square trinomials, or use techniques like factoring by grouping or using the quadratic formula for quadratic factors. Once the factors are found, we can use them to simplify the expression or break it down into partial fractions if the denominator contains repeated or irreducible factors.

Repeated Quadratic Factors

In partial fraction decomposition, when a denominator contains a quadratic factor raised to a power, like \( (x^{2}+7)^{2} \) in our example, it indicates that we are dealing with repeated quadratic factors.

It's essential to handle these carefully: for each repeated factor, we must write a term for every power up to the highest repetition. Each term has its own set of constants, and the numerators are typically formatted as an increasing sequence of polynomial degrees.

Example of Decomposing Repeated Quadratic Factors

In our original expression, we decompose \( (x^{2}+7)^{2} \) into \( \frac{A}{x^{2}+7} +\frac{Bx+C}{(x^{2}+7)^{2}} \). Here, the numerator of the second term is a linear polynomial, following the quadratic factor in the denominator. This pattern ensures that the decomposition is valid for integration or solving.

Algebraic Fractions

Algebraic fractions are fractions that include polynomial expressions. They are the algebraic counterpart to arithmetic fractions, which only contain numbers.

When dealing with algebraic fractions, especially more complicated ones, simplification can vastly improve our ability to work with them. Partial fraction decomposition is one such simplification technique, which is especially useful when the denominators have factors with powers greater than one.

Understanding how to work with algebraic fractions is pivotal in multiple areas of mathematics, not the least of which includes simplifying complex expressions for calculus or solving equations in algebra.