Question

factor the numerator and the denominator of each rational function to determine the locations of any holes or vertical asymptotes. drag each tile to the correct location. each tile can be used more than once, but not all tiles will be used.
rational function | hole location | vertical asymptote location
$f(x) = \frac{x - 1}{x^2 + 4x - 5}$ | |
$f(x) = \frac{x + 2}{x^2 - 2x - 8}$ | |
$f(x) = \frac{6x + 30}{x^2 + 7x + 10}$ | |
$f(x) = \frac{x^2 - 2x}{x - 1}$ | |
tiles: none, $x = -5$, $x = 4$, $x = 1$, $x = -2$, $x = 2$

Explanation:

Step1: Factor $f(x)=\frac{x-1}{x^2+4x-5}$

Factor denominator: $x^2+4x-5=(x+5)(x-1)$
Simplify: $f(x)=\frac{x-1}{(x+5)(x-1)}$, $x
eq1,-5$

Step2: Identify hole/asymptote for $f(x)=\frac{x-1}{x^2+4x-5}$

Common factor: $x-1$, so hole at $x=1$; vertical asymptote at $x=-5$

Step3: Factor $f(x)=\frac{x+2}{x^2-2x-8}$

Factor denominator: $x^2-2x-8=(x-4)(x+2)$
Simplify: $f(x)=\frac{x+2}{(x-4)(x+2)}$, $x
eq-2,4$

Step4: Identify hole/asymptote for $f(x)=\frac{x+2}{x^2-2x-8}$

Common factor: $x+2$, so hole at $x=-2$; vertical asymptote at $x=4$

Step5: Factor $f(x)=\frac{6x+30}{x^2+7x+10}$

Factor numerator/denominator: $6x+30=6(x+5)$, $x^2+7x+10=(x+2)(x+5)$
Simplify: $f(x)=\frac{6(x+5)}{(x+2)(x+5)}$, $x
eq-5,-2$

Step6: Identify hole/asymptote for $f(x)=\frac{6x+30}{x^2+7x+10}$

Common factor: $x+5$, so hole at $x=-5$; vertical asymptote at $x=-2$

Step7: Factor $f(x)=\frac{x^2-2x}{x-1}$

Factor numerator: $x^2-2x=x(x-2)$
Simplify: $f(x)=\frac{x(x-2)}{x-1}$, no common factors

Step8: Identify hole/asymptote for $f(x)=\frac{x^2-2x}{x-1}$

No common factors, so hole: none; vertical asymptote at $x=1$

Answer:

Rational Function	Hole Location	Vertical Asymptote Location
$f(x)=\frac{x+2}{x^2-2x-8}$	$x=-2$	$x=4$
$f(x)=\frac{6x+30}{x^2+7x+10}$	$x=-5$	$x=-2$
$f(x)=\frac{x^2-2x}{x-1}$	none	$x=1$