Question

`(x - 1)/((x + 2)^(2) )`write the partial fraction decomposition.

Answer 1

Explanation

We are given the following expression:

`(x-1)/((x+2)^2)`

We are required to determine the partial fraction decomposition of the given expression.

This is achieved thus:

We know that the partial fraction form of repeated roots is given as:

`(f(x))/((x+a)^2)=(A)/(x+a)+(B)/((x+a)^2)`

Therefore, we have:

`(x-1)/((x+2)^2)=(A)/(x+2)+(B)/((x+2)^2)`

Next, we take the LCD and simplify as follows:

`\begin{gathered} (x-1)/((x+2)^(2))=(A)/(x+2)+(B)/((x+2)^(2)) \\ (x-1)/((x+2)^(2))=(A(x+2)+B)/((x+2)^2) \\ \Rightarrow x-1=A(x+2)+B\text{ ----- \lparen equation 1\rparen} \end{gathered}`

Next, we determine the values of A and B as follows:

`\begin{gathered} x-1=A(x+2)+B \\ \text{ Let x = -2} \\ -2-1=A(-2+2)+B \\ -3=B \\ \therefore B=-3 \\ \\ From\text{ }x-1=A(x+2)+B \\ \text{ Let x = 0} \\ 0-1=A(0+2)+B \\ -1=2A+B \\ \text{ Substitute for B} \\ -1=2A-3 \\ 2A=2 \\ A=1 \end{gathered}`

Therefore, the partial fraction becomes:

`\begin{gathered} (x-1)/((x+2)^2)=(1)/(x+2)+(-3)/((x+2)^2) \\ \Rightarrow(x-1)/((x+2)^2)=(1)/(x+2)-(3)/((x+2)^2) \end{gathered}`

Hence, the answer is:

`\begin{equation*} (1)/(x+2)-(3)/((x+2)^2) \end{equation*}`