Chapter 4.6, Problem 24E

To determine

To decompose: Expression $\frac{5m-4}{m^2-4}$ into partial fractions.

Answer to Problem 24E

  $\frac{5m-4}{\left(m+2\right)\left(m-2\right)}=\frac{7}{2\left(m+2\right)}+\frac{3}{2\left(m-2\right)}$

Explanation of Solution

Given information: $\frac{5m-4}{m^2-4}$

Calculation:

  $\frac{5m-4}{m^2-4}$

Factor denominator

  $\frac{5m-4}{\left(m+2\right)\left(m-2\right)}$

Since the factors are linear, the numerators must be constants.

  $\frac{5m-4}{\left(m+2\right)\left(m-2\right)}=\frac{A}{m+2}+\frac{B}{m-2}$

Multiply both sides by the common denominator.

  $5m-4=A\left(m-2\right)+B\left(m+2\right)$

Letting $m=-2$ gives you:

  $-14=-4A\to A=\frac{7}{2}$

Letting m = 2 give you:

  $6=4B\to B=\frac{3}{2}$

Therefore:

  $\frac{5m-4}{\left(m+2\right)\left(m-2\right)}=\frac{7}{2\left(m+2\right)}+\frac{3}{2\left(m-2\right)}$