Problem 25 - Precalculus: Functions and Graphs

Question

Find the partial fraction decomposition. \(\frac{3 x^{2}-16}{x^{2}-4 x}\)

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{4}{x} - \frac{1}{x-4}\).

Step 1: Factor the Denominator

First, factor the denominator \(x^2 - 4x\). We can factor it as \(x(x - 4)\).

Step 2: Set Up Partial Fraction Decomposition

The next step is to set up the partial fraction decomposition format. Since the denominator factors into \(x(x - 4)\), the partial fraction decomposition will be:\[ \frac{3x^2 - 16}{x(x-4)} = \frac{A}{x} + \frac{B}{x-4} \] where \(A\) and \(B\) are constants that we need to solve for.

Step 3: Clear the Denominator

Multiply through by the common denominator \(x(x-4)\) to clear the fractions:\[ 3x^2 - 16 = A(x - 4) + Bx \]

Step 4: Expand and Collect Like Terms

Expand the equation on the right:\[ 3x^2 - 16 = Ax - 4A + Bx \] Combine like terms:\[ 3x^2 - 16 = (A + B)x - 4A \]

Step 5: Solve for Constants A and B

Now, equate the coefficients from both sides of the equation. Since there is no \(x^2\) term on the right, we get:1. \(A + B = 3\)2. \(-4A = -16\)From equation 2, solve for \(A\):\[ A = 4 \]Substitute \(A = 4\) into equation 1:\[ 4 + B = 3 \]Thus, \(B = -1\).

Step 6: Write the Final Decomposition

Now, substitute \(A\) and \(B\) back into the partial fraction decomposition format:\[ \frac{3x^2 - 16}{x(x-4)} = \frac{4}{x} - \frac{1}{x-4} \]

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex polynomial into simpler products that are easier to work with. This is similar to finding the prime factors of numbers but for polynomials. In the given exercise, the polynomial in the denominator is factored to facilitate partial fraction decomposition. Our polynomial was \(x^2 - 4x\). To factor it, we look for common factors in each term. Here, both terms share \(x\) as a common factor. Thus, we can represent the expression as:\[ x^2 - 4x = x(x - 4) \]. This makes it simpler to handle in further operations like decomposition. Factoring is a crucial step as it simplifies rational expressions, making it easier to solve algebraic equations.

Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator, denominator, or both. They are similar to regular fractions but with expressions involving variables. In partial fraction decomposition, handling these rational expressions becomes more streamlined. The problem involves the rational expression \(\frac{3x^2-16}{x^2-4x}\). To decompose this expression, we first factor the denominator, which helps set the stage for breaking it into simpler fractions or components.

By having a form like \(\frac{A}{x} + \frac{B}{x-4}\), we turn the complex fraction into manageable parts that can be easily integrated or manipulated in other algebraic operations.

This process is essential in calculus for integration and in various algebraic applications where simpler forms of expressions are required.

Algebraic Equations

In dealing with algebraic equations, especially those involving partial fraction decomposition, equating coefficients is a key step. Once the rational expression is split into simpler fractions, as in our case \(\frac{A}{x} + \frac{B}{x-4}\), the original equation can be rewritten to solve for the unknowns \(A\) and \(B\). This involves expanding terms, and clearing fractions by multiplying through by the least common multiple of denominators.

The next critical action is aligning terms by their degree (like terms) and setting up systems of equations. For example, equating the coefficients for each power of \(x\) from the original & expanded polynomials allows us to solve for \(A\) and \(B\). In our problem, it led to: \(A + B = 3\) and \(-4A = -16\). Solving these gives us the values \(A = 4\) and \(B = -1\).

This linear equation method circumvents solving quadratic equations directly, simplifying and neatly solving complex algebraic equations progressively.