Problem Statement
Consider the rational function $f(x) = \frac{x^2 - 3x + 5}{x - 2}$ for $x \ne 2$.
(a) Write $f(x)$ in the form $ax + b + \frac{c}{x - 2}$, where $a, b, c \in \mathbb{Z}$. [3 Marks]
(b) Hence, write down the equations of the vertical asymptote and the oblique (slant) asymptote. [3 Marks]
Verified Solution & Marking Scheme
Part (a): Polynomial Long Division
Divide $x^2 - 3x + 5$ by $x - 2$:
$x^2 - 3x + 5 = (x - 2)(x - 1) + 3$
Verification:
$(x - 2)(x - 1) + 3 = x^2 - 3x + 2 + 3 = x^2 - 3x + 5$
Therefore:
$f(x) = x - 1 + \frac{3}{x - 2}$
Here $a = 1, b = -1, c = 3$.
Part (b): Determine Asymptote Equations
Vertical asymptote occurs at denominator zero: $x - 2 = 0 \implies x = 2$.
As $x \to \pm\infty$, $\frac{3}{x - 2} \to 0$, so the graph approaches the linear quotient:
Oblique asymptote: $y = x - 1$.