Problem 21 - A Graphical Approach to College Algebra

Question

Find the partial fraction decomposition for each rational expression. $$\frac{3 x-2}{(x+4)\left(3 x^{2}+1\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is: \( \frac{-2}{7(x+4)} + \frac{6x - 3}{7(3x^2+1)} \).

Step 1: Identify Partial Fraction Form

First, identify the form of the partial fraction decomposition. The expression \( \frac{3x-2}{(x+4)(3x^2+1)} \) will decompose into \( \frac{A}{x+4} + \frac{Bx+C}{3x^2+1} \), where \( A, B, \) and \( C \) are constants to be determined.

Step 2: Multiply by Denominator

To eliminate the denominators, multiply each term by \((x+4)(3x^2+1)\). The equation becomes:\[ 3x - 2 = A(3x^2+1) + (Bx+C)(x+4) \]

Step 3: Expand Terms

Expand the right-hand side of the equation:\[ A(3x^2 + 1) = 3Ax^2 + A \]\[ (Bx+C)(x+4) = Bx^2 + 4Bx + Cx + 4C \]So, the equation becomes:\[ 3x - 2 = 3Ax^2 + A + Bx^2 + 4Bx + Cx + 4C \]

Step 4: Combine Like Terms

Combine like terms from the expansion:\[ 3x - 2 = (3A + B)x^2 + (4B + C)x + (A + 4C) \]

Step 5: Set Up System of Equations

Equate the coefficients from both sides of the equation:For \( x^2 \): \( 3A + B = 0 \)For \( x \): \( 4B + C = 3 \)For constant terms: \( A + 4C = -2 \)

Step 6: Solve the System of Equations

Solve the system of equations using substitution or elimination:1. Solve \( 3A + B = 0 \) for \( B \): \( B = -3A \).2. Substitute \( B = -3A \) into \( 4B + C = 3 \): \(-12A + C = 3 \Rightarrow C = 3 + 12A \).3. Substitute \( A \) and \( C = 3 + 12A \) into \( A + 4C = -2 \): \[ A + 4(3 + 12A) = -2 \] \[ A + 12 + 48A = -2 \] \[ 49A = -14 \rightarrow A = -\frac{14}{49} = -\frac{2}{7} \].4. Substitute \( A = -\frac{2}{7} \) back into \( B = -3A \) and \( C = 3 + 12A \): \[ B = 3 \times \frac{2}{7} = \frac{6}{7} \] \[ C = 3 - 12 \times \frac{2}{7} = 3 - \frac{24}{7} = 3 - \frac{24}{7} = \frac{21}{7} - \frac{24}{7} = -\frac{3}{7} \].

Step 7: Write Partial Fraction Decomposition

Now substitute back the values of \( A, B, \) and \( C \) into the partial fraction form:\[ \frac{3x-2}{(x+4)(3x^2+1)} = \frac{-\frac{2}{7}}{x+4} + \frac{\frac{6}{7}x - \frac{3}{7}}{3x^2+1} \]Convert into a simpler form:\[ \frac{-2}{7(x+4)} + \frac{6x - 3}{7(3x^2+1)} \]

Key Concepts

Rational Expressions

When we talk about rational expressions, we're looking at fractions where both the numerator and the denominator are polynomials. In the exercise provided, the rational expression is \( \frac{3x-2}{(x+4)(3x^2+1)} \). Understanding the components of this fraction is essential:

• The numerator is \( 3x-2 \), a simple linear polynomial.
• The denominator is \( (x+4)(3x^2+1) \), a product of a linear polynomial and a quadratic polynomial.

The goal of partial fraction decomposition is to break down this complex fraction into simpler fractions that can be handled more easily, especially when finding integrals or solving equations. It can make complex algebraic manipulations much more manageable and help you identify real or complex roots.In this context, partial fraction decomposition involves expressing the given rational expression as a sum of simpler fractions, which will then point us towards possible integrations in terms of simpler forms.

System of Equations

Solving the system of equations is crucial when determining the constants in partial fraction decomposition. After setting up our expression in terms of coefficients, we equate it to form a system of equations:- For \( x^2 \), the equation was \( 3A + B = 0 \).- For \( x \), it was \( 4B + C = 3 \).- For the constant terms, \( A + 4C = -2 \).The task is to find the values of \( A, B, \) and \( C \) by solving these equations. This involves basic manipulation techniques such as substitution or elimination. Substitution helps replace one variable with another known equation, as seen when \( B = -3A \) is used in subsequent equations.Finally, solving these three equations allows us to uniquely determine \( A, B, \) and \( C \). Calculating these values is essential because it lets us construct our decomposed fractions, giving us a clearer, more useful form.

Algebraic Techniques

The partial fraction decomposition process involves various algebraic techniques such as expanding polynomial products and combining like terms. Let's look at how these techniques were applied:

• After restructuring the expression, each term was multiplied to eliminate the denominator, bringing us to equation matching where we have \( 3x - 2 = A(3x^2+1) + (Bx+C)(x+4) \).
• Expanding the terms involved using distributive principles: \( A(3x^2 + 1) \) results in \( 3Ax^2 + A \), and \( (Bx+C)(x+4) \) results in \( Bx^2 + 4Bx + Cx + 4C \).
• Combining like terms simplifies everything, creating a straightforward comparison between coefficients, making it easier to form a system of equations.

Mastering these algebraic steps is key for partial fraction decomposition and other algebraic operations. They are useful in broad mathematical contexts beyond just this exercise.