Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{1}{4 x^{2}-9}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given expression \(\frac{1}{4x^{2}-9}\) is \(-\frac{1}{9(2x + 3)} + \frac{1}{9(2x - 3)}\). This has been verified algebraically and can also be confirmed graphically.

Step 1: Factor the denominator

The given expression is \( \frac{1}{4x^{2}-9} \). The denominator \( 4x^{2}-9 \) is a difference of squares, and can be factored as \( (2x + 3)(2x - 3) \). Thus, the expression becomes: \( \frac{1}{(2x + 3)(2x - 3)} \).

Step 2: Write the expression as a sum of partial fractions

After factoring, the expression can be decomposed into partial fractions. For each factor in the denominator, we write a separate fraction: \( \frac{A}{2x + 3} + \frac{B}{2x - 3} \). The goal is to find the values of A and B that make this equation true.

Step 3: Solve for the constants A and B

To find the values of A and B, we will multiply both sides by \( (2x + 3)(2x - 3) \): \( 1 = A(2x - 3) + B(2x + 3) \). Now solve the equation for two distinct points. For x = -3/2, we get A = -1/9. For x = 3/2, we get B = 1/9. So, the partial fraction decomposition of the given expression is \( -\frac{1}{9(2x + 3)} + \frac{1}{9(2x - 3)} \).

Step 4: Verify the result algebraically

Combine the partial fractions to check if the result equals the original expression. Multiply \( -\frac{1}{9(2x + 3)} + \frac{1}{9(2x - 3)} \) with the least common denominator \( (2x + 3)(2x - 3) \) to simplify the expression. It results in \( 1 = (4x^{2} - 9) \). Thus, the partial fraction decomposition has been verified algebraically.

Step 5: Verify the result graphically

Graph the original function and the decomposed expression on the same graph. If correct, the two will coincide.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. In such expressions, operations like addition, subtraction, multiplication, and division are performed similar to numerical fractions but involve polynomial expressions. Typically, their denominators should not be zero to avoid undefined values. Identifying the domain of rational expressions, i.e., the set of allowable values for the variable, is a crucial step in examining these expressions. Understanding the behavior of rational expressions is fundamental when working with algebraic concepts, such as finding the least common denominator or simplifying these expressions through factorization.

Algebraic Verification

Algebraic verification involves taking the expression or outcome you've derived and checking it to ensure it's mathematically accurate. In partial fraction decomposition, you decompose a complex rational expression into simpler fractions.

• Once decomposed, verify by combining these fractions to see if they equal the original expression.
• Multiply each term by their denominators to combine them back into a single expression.

This step confirms that the values you found for the constants in the partial fractions are correct by ensuring the operation reverses back to the initial given expression. Algebraic verification strengthens the understanding of each component's role in the overall expression.

Graphical Verification

Graphical verification is using a graphing utility or calculator to confirm the results of calculations. Begin by graphing both the original rational expression and the expression resulting from partial fraction decomposition.

• If both graphs coincide and overlap perfectly in the same viewing window, it shows that the decomposition was correct.
• Any deviation suggests potential calculation errors that should be re-examined.

Visual interpretation via graphs helps reinforce the mathematical integrity of your solution, offering a complementary method for double-checking algebraic work.

Factoring Differences of Squares

Factoring differences of squares is a specific technique used in algebra to simplify expressions or equations where there's a difference between two perfect squares. The expression takes the form of \(a^2 - b^2\), and it can be factored into \((a + b)(a - b)\).

• Recognizing opportunities to factor expressions, like \(4x^2 - 9\) into \((2x+3)(2x-3)\), is crucial.
• This technique drastically simplifies and enables us to decompose expressions into partial fractions.

Factoring reduces the complexity of expressions and is an essential step in solving a wide range of algebraic problems.