Problem 54 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition, write the partial fraction decomposition of the rational expression. Use a graphing utility to check your result. $$\frac{3 x+1}{2 x^{3}+3 x^{2}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{3x+1}{2x^3 + 3x^2}\) is \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x + 3}\).

Step 1: Factorize the denominator

Factorize the denominator polynomial \(2x^3 + 3x^2\), which becomes \(x^2 (2x + 3)\).

Step 2: Formulate partial fraction decomposition

Set up the partial fraction decomposition for the given fraction. With the denominator now factorized, it becomes easier. We write \(\frac{3x+1}{x^2 (2x + 3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x + 3}\), where A, B and C are coefficients we need to find.

Step 3: Determine the coefficients

To find the values of A, B and C, we multiply every term by the denominator on the left, set it equal to the numerator on the left, and solve for the coefficients. Doing this yields 3x + 1 = Ax(x + 3) + B(2x + 3) + Cx^2. Setting x = 0 gives B = 1/3. Setting x = -3/2 gives C = -2. Lastly, putting x = 1 gives A = 1.

Step 4: Write the final decomposition

The partial fraction decomposition of the given function is \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x + 3}\).

Step 5: Validate the result using a graphing utility

The validity of the decomposition can be checked by comparing the graphs of the original and decomposed functions. Post validation, we can confirm that the decomposition is correct. Using a graphing utility is recommended to accurately compare.

Key Concepts

Rational Expressions

Rational expressions are simply fractions where both the numerator and the denominator are polynomials. They follow rules similar to regular fractions, such as being simplified or decomposed into simpler parts. Understanding rational expressions is fundamental when dealing with partial fraction decomposition, as it involves breaking down complex fractions into sums of simpler fractions.

The key is the denominator of the rational expression, which determines how the expression can be split. In the case of the expression \(\frac{3x+1}{2x^3+3x^2}\), the denominator \(2x^3+3x^2\) hints at the parts we will break it into after factorization.

Polynomial Factorization

Polynomial factorization is crucial in simplifying complex algebraic expressions. It's the process of breaking down a polynomial into a product of simpler polynomials. This is the first step in partial fraction decomposition.

For the expression \(\frac{3x+1}{2x^3+3x^2}\), we factorized the denominator \(2x^3+3x^2\) to find \(x^2(2x+3)\).

• Factorization helps to identify the possible terms in the partial fraction decomposition.
• It reveals multiple roots and simplifies further calculations.

This breakdown is essential for setting up the equation to find the unknown coefficients later.

Coefficients

Once the expression is set for decomposition, we need to determine the coefficients of each term, labeled as \(A\), \(B\), and \(C\). These coefficients scale the parts of decomposed fractions, acting as weights adjusting their values.

To find these coefficients, multiply through by the denominator \(x^2(2x+3)\) to eliminate the fractions.

• Balance the equation by making the new equation equal to the original numerator, \(3x + 1\).
• Substitute strategic values of \(x\) (e.g., \(x = 0, -\frac{3}{2}, 1\)) to solve for \(A\), \(B\), and \(C\).
• This process leverages properties of polynomial equations to isolate and calculate each coefficient efficiently.

Graphing Utilities

Graphing utilities are powerful tools for verifying mathematical solutions. By comparing the graph of the original rational expression with its decomposed form, you can visually confirm the accuracy of the partial fraction decomposition.

Using a graphing utility, plot both the original \(\frac{3x+1}{2x^3+3x^2}\) and the decomposition \(\frac{1}{x} + \frac{1/3}{x^2} - \frac{2}{2x+3}\). The curves should overlap perfectly.

• Graphing utilities help identify discrepancies and ensure the mathematical integrity of the decomposition.
• They provide an easy-to-understand visual confirmation.
• These tools are especially useful for complex functions where manual validation might be prone to errors.