Problem 27 - A Graphical Approach to Precalculus with Limits

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

\(\frac{-2/7}{x+4} + \frac{(6/7)x + (3/7)}{3x^2 + 1}\)

Step 1: Identify the Denominator Factors

The given rational expression is \(\frac{3x - 2}{(x+4)(3x^2 + 1)}\). The denominator contains two distinct irreducible factors: \(x+4\) (a linear factor) and \(3x^2 + 1\) (a quadratic factor).

Step 2: Set Up Partial Fraction Form

Since the denominator consists of a linear factor \((x+4)\) and a quadratic factor \((3x^2 + 1)\), the partial fraction decomposition can be set up as: $$\frac{3x - 2}{(x+4)(3x^2 + 1)} = \frac{A}{x+4} + \frac{Bx + C}{3x^2 + 1}$$ where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the Denominator

Multiply through by the denominator \((x+4)(3x^2 + 1)\) to clear the fractions: \[ 3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4) \]

Step 4: Expand and Simplify

Expand the right side of the equation:\[ 3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4) = 3Ax^2 + A + (Bx^2 + 4Bx + Cx + 4C) \]Simplify to: \[ 3x - 2 = (3A + B)x^2 + (4B + C)x + (A + 4C) \]

Step 5: Solve for Coefficients

By comparing coefficients with \(3x - 2 = 0x^2 + 3x - 2\), we get:1. \(3A + B = 0\) (for \(x^2\))2. \(4B + C = 3\) (for \(x\))3. \(A + 4C = -2\) (constant term).

Step 6: Solve the System of Equations

From the first equation, \(B = -3A\). Substitute into the other equations:1. \(4(-3A) + C = 3\) gives \(-12A + C = 3\) or \(C = 3 + 12A\).2. \(A + 4(3 + 12A) = -2\) gives \(A + 12 + 48A = -2\) or \(49A = -14\), solving gives \(A = -\frac{2}{7}\).Using \(A = -\frac{2}{7}\), substitute back to find \(B\) and \(C\): \(B = -3(-\frac{2}{7}) = \frac{6}{7}\) and \(C = 3 + 12(-\frac{2}{7}) = \frac{3}{7}\).

Step 7: Write the Partial Fraction Decomposition

Substitute \(A = -\frac{2}{7}\), \(B = \frac{6}{7}\), and \(C = \frac{3}{7}\) back into the partial fractions:\[\frac{3x - 2}{(x+4)(3x^2 + 1)} = \frac{-2/7}{x+4} + \frac{(6/7)x + (3/7)}{3x^2 + 1}\]

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. The expression may look complex, but it’s just a ratio of two polynomial expressions. Decomposing rational expressions into simpler fractions makes them easier to work with, especially in calculus and algebra. By breaking down complicated expressions, you can simplify integration or solve equations with increased accuracy. In the example \[\frac{3x - 2}{(x+4)(3x^2 + 1)}\] this rational expression is composed of a numerator (\(3x - 2\)) and a denominator, which is a product of two factors: \((x + 4)(3x^2 + 1)\). The challenge involves making this expression easier to interpret and solve by using partial fraction decomposition.

Irreducible Factors

Before you can decompose a rational expression, identify its irreducible factors. Irreducible factors are those that cannot be factored further over the real numbers. These factors can be either linear or quadratic. In our case, the denominator consists of

• \(x+4\): a linear factor, and
• \(3x^2 + 1\): a quadratic factor.

The linear factor is straightforward since it’s a simple polynomial of degree one. For the quadratic factor, you should verify if it can be factored further. Since \(3x^2 + 1\) doesn't have real roots (as shown through the discriminant), it is irreducible and must be accounted for in the decomposition process.

System of Equations

Once the partial fraction form is set up, solving for the unknown constants involves creating a system of equations. This step ensures that both sides of the equation are identical once expanded and rearranged. Taking the given expression: \[3x - 2 = A(3x^2 + 1) + (Bx + C)(x + 4)\] we expand and simplify the equation. By equating coefficients of corresponding terms (\(x^2\), \(x\), and constant terms), we develop a system of equations:

• \(3A + B = 0\) (for \(x^2\))
• \(4B + C = 3\) (for \(x\))
• \(A + 4C = -2\) (constant term)

These equations allow you to solve for constants \(A\), \(B\), and \(C\) essential to write the decomposed form of the rational expression.

Coefficients

Coefficients are the numerical factors that multiply the variables in a polynomial. In partial fraction decomposition, accurately determining coefficients is crucial for properly expressing the original rational expression as a sum of simpler fractions. In our example, after setting up the partial fractions:\[\frac{3x - 2}{(x+4)(3x^2 + 1)} = \frac{A}{x+4} + \frac{Bx + C}{3x^2 + 1}\] solving the system of equations provides the values:

• \(A = -\frac{2}{7}\)
• \(B = \frac{6}{7}\)
• \(C = \frac{3}{7}\)

These coefficients are used to write the decomposed form, making calculations or integrations simpler. Each coefficient aligns with a term of \(x\) in the decomposed parts, linking back to the original fraction equation. The successful calculation and application of these coefficients provide a clearer path to simplifying complex mathematical problems.