Problem 4 - 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. $$\frac{3 x+16}{(x+1)(x-2)^{2}}$$

Step-by-Step Solution

Answer

The form of the partial fraction decomposition of \( \frac{3 x+16}{(x+1)(x-2)^{2}} \) is \( \frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^{2}} \)

Step 1: Identify terms in denominator

The denominator of the given rational function consists of two terms: \((x+1)\) and \((x-2)^{2}\). Let's consider them as separate fractions in the form of partial fractions.

Step 2: Writing the form of partial fraction decomposition

The general form of the partial fractions decomposition will be: \[ \frac{3 x+16}{(x+1)(x-2)^{2}} = \frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^{2}} \] where A, B, and C are constants.

Key Concepts

Rational Expressions

Rational expressions are mathematical quotients consisting of a numerator and a denominator, both of which are polynomials. Just like fractions, rational expressions can be added, subtracted, multiplied, and divided. One unique feature of rational expressions is that their values are only defined where the denominator does not equal zero. This condition is crucial because denominators that equal zero lead to undefined expressions.

In the given exercise, we have the rational expression \( \frac{3x+16}{(x+1)(x-2)^2} \). The numerator here is \( 3x + 16 \) and the denominator is \( (x+1)(x-2)^2 \). We can manipulate this rational expression using partial fraction decomposition, which simplifies the expression into a sum of simpler fractions. This procedure makes more complex algebraic operations easier to handle.

Denominator Terms

Identifying the terms of the denominator in a rational expression is a key step in partial fraction decomposition. The denominator comprises polynomial factors that guide the form of the decomposition. Each different factor results in a separate term in the partial fraction decomposition.

For instance, the denominator \( (x+1)(x-2)^2 \) of our expression consists of two distinct factors: \((x+1)\) and \((x-2)\). The factor \((x-2)\) appears with a power of two, indicating that it is a repeated factor. For such a repeated factor, we include multiple terms in the decomposition to account for both the simple factor and higher powers. This results in individual terms \( \frac{B}{x-2} \) and \( \frac{C}{(x-2)^2} \) for the decomposition, reflecting both the simple factor and its square.

Constants in Partial Fractions

In partial fraction decomposition, unknown constants are used to represent the numerators of the simpler fractions that form the decomposition. These constants (usually denoted as \( A, B, \) and \( C \) in our instance) need to be determined through further calculation when solving the decomposition fully.

For the expression \( \frac{3x+16}{(x+1)(x-2)^2} \), the form involves dividing the expression into simpler fractions: \( \frac{A}{x+1} \), \( \frac{B}{x-2} \), and \( \frac{C}{(x-2)^2} \). Each of these fractions has a constant numerator which represents an unknown value to be solved for later. By arranging these constants correctly, the decomposed fractions accurately sum up to form the original rational expression.

• \( A \) is the constant for the term with the denominator \((x+1)\).
• \( B \) corresponds to the regular power of \((x-2)\).
• \( C \) is used for the higher power, \((x-2)^2\).

Understanding these constants is essential for reconstructing the rational expression from its components or to simplify complex expressions for further calculations.