Problem 56 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x^{2}-3 x+2}{4 x^{3}+11 x^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression is \(\frac{A}{x^{2}} + \frac{B}{4x+11}\).

Step 1: Factorize the denominator

To factor the denominator \(4x^{3}+11x^{2}\), it is noticeable that both terms have a common factor of \(x^{2}\). So, factor out \(x^{2}\) from both terms to obtain \(x^{2}(4x + 11)\).

Step 2: Write the form of the partial fractions

The decomposed form of the rational expression will be the sum of two fractions, where each denominator is each of the factors of the original denominator. Therefore, this rational expression can be decomposed into \(\frac{A}{x^2} + \frac{B}{4x+11}\), where A and B are constants.

Key Concepts

Rational Expressions

Rational expressions are a type of mathematical expression that can be thought of as the fraction of two polynomials. The numerator and the denominator are both polynomials, just like fractions you encounter with integers. Understanding these expressions is crucial in algebra as they often pop up in equations and are central to many algebraic techniques.

• The numerator is the polynomial above the fraction line.
• The denominator is the polynomial below the fraction line.

A key characteristic of rational expressions is that the denominator cannot be zero. Having a zero denominator will make the expression undefined.
Working with rational expressions requires ensuring they are simplified. Simplification often involves factoring and canceling out terms, similar to simplifying numerical fractions. This simplification process can make other operations, like addition and subtraction, easier to perform. Whether you aim to solve systems of equations or to understand polynomial division, rational expressions provide the foundation for more comprehensive algebraic tasks.

Algebraic Factoring

Algebraic factoring is a vital skill in simplifying mathematical expressions. It involves breaking down a polynomial into simpler, 'factorable' components. These components, when multiplied together, reconstruct the original polynomial. Factoring is particularly handy when dealing with quadratic equations or complex rational expressions.
First, always check for a Greatest Common Factor (GCF). This is the largest factor that divides each term of the polynomial. In our provided example, the GCF is easily visible as it is common to both terms.

• Factor out the GCF first.
• Look for identifiable patterns, such as a difference of squares or a trinomial square.

By effectively using factoring, you can transform complex expressions into more manageable pieces. This aids in further operations, like simplifying rational expressions or solving equations. Often, factoring is the first step when working on partial fraction decomposition since it reveals possible denominators.

Denominator Factoring

Denominator factoring is a subset of algebraic factoring focused on the polynomial in the fraction's denominator. It's essential for partial fraction decomposition, as breaking down the denominator into simpler terms allows for splitting the entire rational expression into simpler fractions.
In our example, the denominator was initially given as a cubic polynomial. Through factoring, it was broken into parts: first identifying a common factor of \(x^2\), and then revealing a linear term \(4x + 11\).

• Identify any obvious factors first, such as common terms.
• If necessary, factor further into linear or irreducible quadratic terms.

The outcome of denominator factoring not only simplifies the original expression but also helps identify partial fractions. By dealing with simpler components, solving equations, performing integration, or other algebraic processes become more straightforward. Mastery of this technique is a stepping stone to handling more significant mathematical challenges efficiently.