Problem 18 - Algebra and Trigonometry

Question

write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the expression is \(-\frac{3}{x}-\frac{5}{x+1}+\frac{2}{x-5}\).

Step 1: Setting Up the Expression

The first step in the partial fraction decomposition is to set the given rational function equal to a series of fractions, the numerators of which are constants and the denominators of which are the factors of the original denominator. Hence, \( \frac{4x^{2}-5x-15}{x(x+1)(x-5)} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5} \).

Step 2: Multiplying Through by the Common Denominator

Next, we have to clear the fractions, by multiplying both sides of the last equation by the common denominator \(x(x+1)(x-5)\) to get: \( 4x^{2} - 5x - 15 = A(x+1)(x-5) + Bx(x-5) + Cx(x+1) \).

Step 3: Isolate Coefficients and Solve

Next, equate the coefficients of the like terms on both sides of the equation and solve for \(A\), \(B\), and \(C\). This can be done by setting \(x = 0\), \(x = -1\), and \(x = 5\) one by one to solve for \(A\), \(B\), and \(C\) respectively. Doing this we get \(A = -3\), \(B = -5\), and \(C=2\).

Step 4: Writing the Final Answer

Now that we know the constants \(A\), \(B\), and \(C\), plug them back into our original fractions to get the partial fraction decomposition. Therefore, the result will be: \(-\frac{3}{x}-\frac{5}{x+1}+\frac{2}{x-5}\).

Key Concepts

Rational Expressions

Rational expressions are fractions that include polynomials in the numerator, the denominator, or both. Just like fractions in arithmetic which are composed of integers, rational expressions signify division. However, in algebra, the complexity increases as we deal with polynomials.

An essential skill when working with rational expressions is the ability to perform partial fraction decomposition. This method is particularly useful for integrating rational functions or for simplifying complex expressions into more manageable pieces. A rational expression like \(\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}\) can present quite a challenge, but breaking it down into simpler parts helps to make sense of it and work with it more easily.

Identifying the Denominator's Factors

First, recognize that the denominator of a rational expression can be factored into simpler expressions. In this case, the factors are \(x\), \(x+1\), and \(x-5\). These factors lay the groundwork for the decomposition as they will comprise the denominators of the resulting simpler fractions.

Algebraic Techniques

Algebraic techniques involve the array of strategies and procedures used to manipulate algebraic expressions and equations. In partial fraction decomposition, these techniques are pivotal.

Clearing the Fractions

One key technique involves clearing the fractions by multiplying throughout by the common denominator, transforming a complex rational expression into a polynomial equation. This step facilitates the isolation of variables or, in this decomposition, the coefficients \(A\), \(B\), and \(C\).

Equating Coefficients

After clearing the fractions, we equate coefficients of the resulting polynomial to those of the original expression. This part of the process requires careful expansion and simplification. Coefficients of the expanded form are often found by comparing the corresponding terms on both sides of the equation. Through this technique, we directly identify the values needed to complete the decomposition.

Solving Equations

Solving equations is fundamental in algebra. In the context of partial fraction decomposition, we often deal with linear equations when finding the values for \(A\), \(B\), and \(C\). Using strategic values for \(x\) that correlate to the factors of the denominator simplifies the equation and makes the constants apparent.

For instance, setting \(x\) equal to zeros of the denominator will eliminate terms and isolate the desired coefficient—what is also known as the 'cover-up' method. By choosing \(x = 0\), \(x = -1\), and \(x = 5\), we directly solve for \(A\), \(B\), and \(C\) respectively. Solving these equations is typically straightforward, but it's crucial to work systematically to avoid errors and to ensure that each constant corresponds to the correct part of the decomposed expression.

Mastering these algebraic techniques not only aids in partial fraction decomposition but also enhances general problem-solving skills across different areas of mathematics.