Question

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

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

Short answer

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

Step-by-step solution

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,

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

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 Aand B ,we get

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