Partial Fractions - Questions
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Express \(\dfrac{{x + 2}}{{\left( {x - 1} \right)\left( {x - 2} \right)}}\) as a partial fraction
$$
\begin{aligned}
\frac{x+2}{(x-1)(x-2)} &=\frac{A}{x-1}+\frac{B}{x-2} \\ \therefore x+2 &\equiv A(x-2)+B(x-1) \\
& \equiv A x-2 A+B x-B \\
x+2 &\equiv(A+B) x-2 A-B \\
\therefore A+B &=1\quad \color{red}\ldots \text { (1) } \\
-2 A-B &=2 \quad \color{red}\cdots(2)\\
{\color{red}(1)}+{\color{red}(2)}\quad -A &=3 \\
A &=-3 \\
\text{ln}\ {\color{red}(1)}\quad -3+B &=1 \\ B &=4 \\ \therefore\ \frac{x+2}{(x-1)(x-2)} &=\frac{4}{x-2}-\frac{3}{x-1}
\end{aligned}
$$
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