Problem 7 - Algebra and Trigonometry

Question

write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{x^{3}+x^{2}}{\left(x^{2}+4\right)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\) is \(\frac{Ax+B}{x^{2}+4}+ \frac{Cx+D}{(x^{2}+4)^{2}}\).

Step 1: Identify the Type of Factors in the Denominator

The denominator here, \((x^{2}+4)^{2}\), is an irreducible quadratic factor since it cannot be factored further into linear terms or simplified further.

Step 2: Write the Form of Partial Fraction Decomposition

In this case, the general form of a partial fraction decomposition involves a term with just the quadratic factor and another term with the square of the quadratic factor. The general form is \(\frac{Ax+B}{x^{2}+4}+ \frac{Cx+D}{(x^{2}+4)^{2}}\), where A, B, C and D are constants to be determined if it were necessary.

Key Concepts

Irreducible Quadratic Factor

When dealing with algebraic fractions, especially during partial fraction decomposition, the term \textbf{irreducible quadratic factor} often emerges. This wording might appear daunting, but it essentially refers to a quadratic expression that cannot be factored further into real linear elements. This means no real number solutions exist for the equation if you tried to set the quadratic expression to zero.

An irreducible quadratic factor comes into play in partial fraction decomposition because it signifies the 'pieces' that a complex fraction is broken down into will include a term with the quadratic in the denominator. For example, for the expression \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\), the denominator \( (x^{2}+4)^{2} \) is such an irreducible quadratic factor, because \(x^{2}+4\) has no real roots and cannot be factored into simpler linear terms.

Understanding and identifying these factors is crucial in simplifying complex algebraic expressions. It places demands on which form the partial fraction decomposition will adopt, as recognizing the nature of the denominator's factors determines the structure needed for the decomposition.

Rational Expressions

A \textbf{rational expression} is a fraction where both the numerator and the denominator are polynomials. In mathematics, they are akin to the rational numbers, where we have integers in the numerator and denominator. The function \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\) from our exercise is an example of a rational expression.

Just like rational numbers, rational expressions can often be simplified. To do this, one might factor the polynomials and cancel any common factors. However, some rational expressions contain irreducible quadratic factors, as seen in the exercise, which means their simplification process includes partial fraction decomposition—breaking down a complex fraction into simpler 'partial' fractions.

In addition, solving for variables in equations involving rational expressions often requires that these expressions be rewritten in simpler forms. Thus, understanding how to handle rational expressions is a fundamental block of algebra that enables the solving of complex equations.

Algebraic Fractions

The concept of \textbf{algebraic fractions} is closely linked to rational expressions since any fraction featuring polynomials is considered an algebraic fraction. \(\frac{x^{3}+x^{2}}{(x^{2}+4)^{2}}\) can thus be labeled as an algebraic fraction as well. It contains variables, constants, and the operations of addition, subtraction, and exponentiation to whole numbers.

One of the essential skills when working with algebraic fractions is simplifying them. Simplification may involve factoring polynomials, canceling common terms, or employing more advanced techniques like partial fraction decomposition, particularly when there are irreducible quadratic factors in the denominator. The decomposition is a way to disassemble a complicated algebraic fraction into basic components that are easier to integrate or differentiate if one is dealing with calculus, or to evaluate, if the goal is to simplify the expression for other purposes.

It's of great importance to handle algebraic fractions with care, always paying attention to the domain of the expression by checking for values that might make the denominator zero. These values need to be excluded since division by zero is undefined in mathematics. Additionally, manipulating these expressions efficiently paves the way for tackling more specialized math problems, where algebraic fractions play significant roles.