Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Write the partial fraction decomposition for the expression. $$ \frac{8 x+3}{x^{2}-3 x} $$

Step-by-Step Solution

Answer

\(\frac{8x+3}{x(x - 3)} = \frac{-1}{x} + \frac{9}{x - 3}\)

Step 1: Factorize the Denominator

The denominator of the given expression is \(x^{2}-3x\). This can be factorized by taking x common. So, \(x^{2}-3x = x(x - 3)\)

Step 2: Partial Expression

Next, we write the given expression \(\frac{8x+3}{x(x - 3)}\) as the sum of two expressions with the factors of the denominator as denominators: \(\frac{8x+3}{x(x - 3)} = \frac{A}{x} + \frac{B}{x - 3}\)

Step 3: Clear the Fractions

Clearing the fractions, we get \(8x+3 = A(x - 3) + Bx\)

Step 4: Equating the Coefficients

Here, A and B are constants to be determined. We can find their values by equating the coefficients of the like terms on both sides of the equation. Equating the coefficients for x, we get: 8 = A + B. The constant terms should also be equal: 3 = -3A or A = -1. Plugging this into the equation for x we get: 8 = -1 + B, thus B = 9.

Step 5: The Partial Fraction Decomposition

The constants A and B are inserted back into the equation from step 2 giving the result: \(\frac{8x+3}{x(x - 3)} = \frac{-1}{x} + \frac{9}{x - 3}\)