Problem 13 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{7 x-4}{x^{2}-x-12}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given expression is: \(\frac{1}{x-4} + \frac{2}{x+3}\).

Step 1: Factor the denominator

Firstly, factor the denominator of the fraction, which is \(x^{2}-x-12\). Upon factoring, we get \((x-4)(x+3)\).So, we can write the given fraction as: \(\frac{7x-4}{(x-4)(x+3)}\).\n

Step 2: Set up the partial fractions

A general pattern that can be used to break down the fraction is based on the factored denominator. We would assume the fraction can be broken down as: \(\frac{7x-4}{(x-4)(x+3)} = \frac{A}{x-4} + \frac{B}{x+3}\), where \(A\) and \(B\) are constants that we need to solve for.\n

Step 3: Equate coefficients

To determine the values of \(A\) and \(B\), multiply both sides of the equation by \((x - 4)(x + 3)\) to clear the fractions: \(7x - 4 = A(x+3)+ B(x-4)\).\n

Step 4: Solve for A and B

This is a linear equation system. By choosing convenient values of \(x\), we can solve for \(A\) and \(B\). \nChoose \(x = -3\) to solve for \(A\): \(7(-3) - 4 = A(-3+3) + 3B \Longrightarrow A = 1 \n Now choose \(x = 4\) to solve for \(B\), \(7(4) - 4 = 4A + B(4+3) \Longrightarrow B = 2\). \n