Q. 24 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $24$ and $25$, write the partial fraction decomposition of each rational expression.

$\frac{3x+7}{(x+3{)}^2}$

Step-by-Step Solution

Answer

The decomposition fraction is $\frac{3x+7}{(x+3{)}^2}=\frac{3}{x+3}-\frac{2}{(x+3{)}^2}$

Step 1. Given information

An expression is given as $\frac{3x+7}{(x+3{)}^2}$

Step 2. Partial fraction decomposition

Decompose the rational fraction as,

$\frac{3x+7}{(x+3{)}^2}=\frac{A}{x+3}+\frac{B}{(x+3{)}^2}$

Now simplify,

$$\begin{gathered} 3 x+7=A(x+3)+B \\ =A x+B+3 A \end{gathered}$$

Now, this is an identity. We equate the coefficients in power of x.

$\left\{\begin{array}{l}3=A\\ 7=3A+B\end{array}\right.$

Now from above set we conclude that $A=3 \& B=-2$

After substituting the values of A and B , we get

$\frac{3x+7}{(x+3{)}^2}=\frac{3}{x+3}-\frac{2}{(x+3{)}^2}$