Problem 8 - A Graphical Approach to College Algebra

Question

Find the partial fraction decomposition for each rational expression. $$\frac{2}{x^{2}(x+3)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{-\frac{2}{9}}{x} + \frac{\frac{2}{3}}{x^2} + \frac{\frac{2}{9}}{x+3} \).

Step 1: Identify the Denominator

The first step is identifying the factors in the denominator of the rational expression. For \( \frac{2}{x^{2}(x+3)} \), the factors are \( x^2 \) and \( x + 3 \).

Step 2: Set Up Partial Fraction Decomposition Form

Write the partial fraction decomposition set-up based on the identified factors. We have: \[ \frac{2}{x^2(x+3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+3} \] Here, \( A \), \( B \), and \( C \) are constants to be determined.

Step 3: Clear Denominator and Expand Equation

Multiply the entire equation by \( x^2(x+3) \) to eliminate the denominator: \[ 2 = A(x)(x+3) + B(x+3) + C(x^2) \] Expand the right-hand side: \[ 2 = Ax^2 + 3Ax + Bx + 3B + Cx^2 \]

Step 4: Combine Like Terms

Combine like terms from the expanded equation: \[ 2 = (A + C)x^2 + (3A + B)x + 3B \]

Step 5: Equate Coefficients

From \( 2 = (A + C)x^2 + (3A + B)x + 3B \), match coefficients:1. \( A + C = 0 \) (coefficient of \( x^2 \))2. \( 3A + B = 0 \) (coefficient of \( x \))3. \( 3B = 2 \) (constant term)

Step 6: Solve for Constants

Solve the system of equations derived in the previous step:- From \( 3B = 2 \), solve for \( B \): \( B = \frac{2}{3} \).- Substitute \( B = \frac{2}{3} \) into \( 3A + B = 0 \), so \( 3A + \frac{2}{3} = 0 \) gives \( 3A = -\frac{2}{3} \), thus \( A = -\frac{2}{9} \).- Substitute \( A = -\frac{2}{9} \) into \( A + C = 0 \), so \( -\frac{2}{9} + C = 0 \) gives \( C = \frac{2}{9} \).

Step 7: Write the Final Decomposed Expression

Insert the solved constants into the decomposition form:\[ \frac{2}{x^2(x+3)} = \frac{-\frac{2}{9}}{x} + \frac{\frac{2}{3}}{x^2} + \frac{\frac{2}{9}}{x+3} \]

Key Concepts

Rational Expression

A rational expression is a fraction that has polynomials as both its numerator and denominator. Think of it like a regular fraction, but instead of integers, we use polynomial expressions—these might include variables, coefficients, and various algebraic terms.

When dealing with rational expressions, it's crucial to identify the roots of the polynomial in the denominator because they represent the values that would make the expression undefined (as division by zero is not allowed).

To simplify or manipulate such expressions, especially when performing operations like addition or subtraction with other rational expressions, understanding the factorization of the denominators is key. This often involves techniques like partial fraction decomposition that break down a complex rational expression into simpler parts for easier analysis or integration.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, we perform operations similar to arithmetic but with a broader scope, involving both constants and variables.

One of the primary uses of algebra is solving equations and inequalities. This includes finding unknowns and expressing complex ideas simply. When applied to polynomial equations and rational expressions, algebra allows us to solve each expression by setting up equations and solving them systematically.

• Identify and combine like terms.
• Use symbols to represent unknown quantities.
• Apply mathematical operations to both simplify equations and solve for unknown values.

These algebraic principles are employed in partial fraction decomposition to break expressions into manageable pieces that can be more easily manipulated or integrated.

Polynomial Equations

Polynomial equations are equations involving polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is defined by the highest power present in the expression.

For instance, the equation \( x^2 + 3x + 2 \) is a second-degree polynomial (quadratic) because its highest power is 2. Solving polynomial equations involves finding the roots, or solutions, which make the polynomial equal to zero.

In the context of partial fraction decomposition, polynomial equations are key for identifying how each part of the rational expression can be decomposed. By understanding the structure of the polynomial in the denominator, we can determine the appropriate decomposition form and solve for the constants, transforming a complex rational expression into a series of simpler fractions.