Question

identifying holes in rational functions
given the rational function $f(x) = \frac{x^2 + 7x - 18}{x^2 + x - 2}$ find any hole(s) of the function.
$\circ x = 1$
$\circ x = -2$
$\circ x = 2, x = -9$
$\circ x = -9$
$\circ x = -2, x = 1$
$\circ$ there are no holes

Explanation:

Step1: Factor Numerator and Denominator

Factor \(x^2 + 7x - 18\): Find two numbers that multiply to \(-18\) and add to \(7\). The numbers are \(9\) and \(-2\). So, \(x^2 + 7x - 18=(x + 9)(x - 2)\).
Factor \(x^2 + x - 2\): Find two numbers that multiply to \(-2\) and add to \(1\). The numbers are \(2\) and \(-1\). So, \(x^2 + x - 2=(x + 2)(x - 1)\).
Now, \(f(x)=\frac{(x + 9)(x - 2)}{(x + 2)(x - 1)}\).

Step2: Check for Common Factors

A hole occurs when there is a common factor in the numerator and denominator. Here, the numerator factors are \((x + 9)\) and \((x - 2)\), and the denominator factors are \((x + 2)\) and \((x - 1)\). There are no common factors.

Answer:

There are no holes