Partial Fractions — Question 8

Question 8 – Solution

Step 1: Start with the decomposition. $$\frac{1}{x(x-2)(x+3)}=\frac A x+\frac B{x-2}+\frac C{x+3}$$ Step 2: Multiply every term by $x(x-2)(x+3)$. $$1=A(x-2)(x+3)+Bx(x+3)+Cx(x-2).$$ Step 3: Use values that eliminate two terms at a time. $$\begin{align*} x=0:&\quad 1=-6A &&\Longrightarrow& A&=-\frac16,\\ x=2:&\quad 1=10B &&\Longrightarrow& B&=\frac1{10},\\ x=-3:&\quad 1=15C &&\Longrightarrow& C&=\frac1{15}. \end{align*}$$ Step 4: State the ordered triple. $$\boxed{\displaystyle (A,B,C)=\left(-\frac16,\frac1{10},\frac1{15}\right)}.$$