Question

find all holes of the following function. if a hole exists, write your answer as a coordinate point in simplest form.
$f(x)=\dfrac{3x^2 + 24x}{4x + 32}$

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(3x^2 + 24x\). We can factor out a \(3x\) from the numerator: \(3x(x + 8)\).

Next, factor the denominator \(4x + 32\). We can factor out a \(4\) from the denominator: \(4(x + 8)\).

So the function \(f(x)\) becomes:
\[
f(x)=\frac{3x(x + 8)}{4(x + 8)}
\]

Step2: Cancel common factors

We notice that \((x + 8)\) is a common factor in both the numerator and the denominator (as long as \(x
eq - 8\), because if \(x=-8\), the original function is undefined). So we can cancel out the \((x + 8)\) terms:
\[
f(x)=\frac{3x}{4}, \quad x
eq - 8
\]

Step3: Find the hole

A hole in a rational function occurs at the value of \(x\) that makes the canceled factor equal to zero. We set \(x + 8=0\), which gives \(x=-8\).

To find the \(y\)-coordinate of the hole, we substitute \(x = - 8\) into the simplified function \(\frac{3x}{4}\).

Substitute \(x=-8\) into \(\frac{3x}{4}\):
\[
\frac{3(-8)}{4}=\frac{-24}{4}=-6
\]

Answer:

The hole is at the coordinate point \((-8, - 6)\)