Chapter 4.6, Problem 23E

To determine

To decompose: Expression $\frac{x-6}{x^2-2x}$ into partial fractions.

Answer to Problem 23E

  $\frac{x-6}{x\left(x-2\right)}=\frac{3}{x}-\frac{2}{x-2}$

Explanation of Solution

Given information: $\frac{x-6}{x^2-2x}$

Calculation:

  $\frac{x-6}{x^2-2x}$

Factor denominator

  $\frac{x-6}{x\left(x-2\right)}=\frac{A}{x}+\frac{B}{x-2}$

Multiply both sides by common denominator: $x\left(x-2\right)$

  $x-6=A\left(x-2\right)+B\left(x\right)$

Letting x = 0 gives you:

  $-6=-2A\to A=3$

Letting x = 2 give you:

  $-4=2B\to B=-2$

Therefore,

  $\frac{x-6}{x\left(x-2\right)}=\frac{3}{x}-\frac{2}{x-2}$