Problem 18 - College Algebra

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 given function is \(\frac{-5}{x} + \frac{3}{x+1} + \frac{-2}{x-5}\).

Step 1: Formulate the Decomposition

To write the partial fraction decomposition, we first need to identify the denominators of the fractions. In our rational expression, the denominator factors are \(x\), \(x+1\), and \(x-5\). So, we write our fraction as the sum of simpler fractions with these denominators. Let's represent the numerators of these simpler fractions (the coefficients we want to find) as \(A\), \(B\), and \(C\) respectively. So, the decomposition will look like: \[\frac{4x^2 - 5x - 15}{x(x+1)(x-5)} = \frac{A}{x} + \frac{B}{x+1} + \frac{C}{x-5}\]

Step 2: Clear Fractions and Rearrange

This rearrangement will help us solve for coefficients \(A\), \(B\), and \(C\). Multiply the entire equation by the common denominator \(x(x+1)(x-5)\) to clear the fractions. This simplification results in: \[4x^2 - 5x - 15 = Ax(x+1)(x-5) + Bx(x)(x-5) + Cx(x)(x+1)\]

Step 3: Distribute Terms and Group Like Terms

Fagine you are distributing each term on the right side. Once all terms are distributed, collect and group like terms. Our rearranged expression looks like:\[4x^2 - 5x - 15 = -5Ax + 5Bx - Ax^2 - Bx^2 + Ax + Bx - Cx^2 - 5Cx\]

Step 4: Compare Coefficients and Set up Equations

In order to solve for \(A\), \(B\), and \(C\), we need to set up three separate equations by comparing the coefficients of like terms on both sides of the equation. Here are the equations we get from comparing coefficients: \[Equation 1: -A - B - C = 4\]\[Equation 2: -5A + 5B + A + B - 5C = -5\]\[Equation 3: A + B - C = -15\]

Step 5: Solve the Simultaneous Equations

By solving these simultaneous equations, we find that the values of \(A\), \(B\), and \(C\) are -5, 3, and -2 respectively.

Step 6: Write Out the Decomposed Fraction

Finally, we substitute the values for \(A\), \(B\), \(C\) back into our decomposition equation. The partial fraction decomposition of the given rational function is:\[\frac{4x^2 - 5x - 15}{x(x+1)(x-5)} = \frac{-5}{x} + \frac{3}{x+1} + \frac{-2}{x-5}\]