Problem 14 - Precalculus with Limits

Question

Writing the Form of the Decomposition. Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{6 x+5}{(x+2)^{4}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of the given rational expression is \(\frac{A}{(x+2)} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3} + \frac{D}{(x+2)^4}\).

Step 1: Identify the given rational expression.

The given rational expression is \(\frac{6x+5}{(x+2)^{4}}\).

Step 2: Identify the form of each partial fraction.

The partial fraction can be in two forms when the denominator is a power of a linear factor. These forms are \(\frac{A}{(x+2)}\), \(\frac{B}{(x+2)^2}\), \(\frac{C}{(x+2)^3}\) and \(\frac{D}{(x+2)^4}\) where A, B, C, and D are constants.

Step 3: Combine the partial fractions into a single fraction.

The single fraction with the form of the partial fractions is \(\frac{A}{(x+2)} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3} + \frac{D}{(x+2)^4}\) .

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator consist of polynomials. These can appear simple, like \(\frac{1}{x}\), or more complex as seen in exercises such as \(\frac{6x+5}{(x+2)^{4}}\). When handling rational expressions, it's crucial to ensure that the denominator does not equal zero, as this would make the expression undefined.
In mathematics, rational expressions are critical for factoring, solving equations, and functions analysis. Knowing how to work with these expressions is essential, especially when proceeding to more advanced topics like limits and calculus.
The expression given in the exercise, \(\frac{6x+5}{(x+2)^{4}}\), is an example where the power of a polynomial appears in the denominator. This feature makes the expression a perfect candidate for partial fraction decomposition, which simplifies understanding and solving such mathematical expressions.

Linear Factors

Linear factors in a polynomial are expressions of the form \(x + a\). In the example \(\frac{6x+5}{(x+2)^{4}}\), the linear factor is \(x+2\). These are termed 'linear' because they have the highest exponent of one, unlike quadratic or cubic expressions.
When the denominator of a rational expression consists of linear factors raised to a power, the process of partial fraction decomposition becomes systematic. Each power of the linear factor \((x+2)\) in \(\frac{6x+5}{(x+2)^{4}}\) results in distinct terms in the decomposed form.
Understanding the role of linear factors helps break down complex expressions into simpler parts. It is the presence of such factors in the denominator that determines how many separate terms we will have in our partial fraction decomposition. Recognizing these factors and correctly addressing them is key to simplifying rational expressions and solving equations.

Constants in Partial Fractions

In partial fraction decomposition, constants play a crucial role in simplifying rational expressions. These constants, typically denoted as \(A, B, C, ...\), are unknown coefficients that you'll determine by setting up equations from each part of the decomposition.
For the given expression \(\frac{6x+5}{(x+2)^{4}}\), the partial fraction decomposition involves constants \(A, B, C,\) and \(D\) for each power of the linear factor \((x+2)\). When performing decomposition, your primary task is to express \(\frac{6x+5}{(x+2)^{4}}\) as \(\frac{A}{(x+2)} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3} + \frac{D}{(x+2)^4}\).
Setting up this form is just the beginning. Solving for these constants typically involves substituting values and using algebraic techniques such as equating coefficients. Mastery of identifying and solving for these constants enables one to transform complex rational expressions into simpler, more workable forms.