Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the expression. $$ \frac{3 x-4}{(x-5)^{2}} $$

Step-by-Step Solution

Answer

The partial fraction decomposition of \(\frac{3x - 4}{(x - 5)^2}\) is \(\frac{1/3}{x - 5} + \frac{-8/3}{(x - 5)^2}\).

Step 1: Identify the Form for Partial Fractions

In the expression \(\frac{3x-4}{(x-5)^2}\), the denominator is a repeating factor. Thus, the partial fraction decomposition is of the form:\[ \frac{A}{x-5} + \frac{B}{(x-5)^2} \] Where A and B are constants to be determined.

Step 2: Set Up Equation

Multiply both sides of the equation by the common denominator \((x-5)^2\) to eliminate the fractions:\[ 3x - 4 = A(x - 5) + B \]

Step 3: Solve for Constants (Part I)

For any value of x except x = 5, the equation must hold true. Choose an x value, other than 5, to make the equation easier to solve. Let x = 0, then the equation becomes:\[ -4 = -5A + B \] This gives us our first equation.

Step 4: Solve for Constants (Part II)

Choose another x value. Let x = 1, then the equation becomes:\[ -1 = -4A + B\]This gives us the second equation.

Step 5: Solve The System of Equations

Solve the system of equations:\[ {-4 = -5A + B}, {-1 = -4A + B}\]From these, we get A = 1/3 and B = -8/3.

Step 6: Write Out the Decomposition

Substitute A and B into the partial fraction decomposition form:\[ \frac{3x - 4}{(x - 5)^2 }= \frac{1/3}{x - 5} + \frac{-8/3}{(x - 5)^2}\]This is the partial fraction decomposition of the given rational function.