Question

Determine whether (x) = x² − 2x − 3/x² + 3x + 2 has any holes. If it does, give the coordinates.

Options:
Option 1: (0, 3)
Option 2: (1, -1)
Option 3: (2, 3)
Option 4: (3, 5)

Answer 1

The function (x) = x² − 2x − 3/x² + 3x + 2 has a hole at x = 3.

Step-by-step explanation:

To determine if the function (x) = x² − 2x − 3/x² + 3x + 2 has any holes, we need to check if there are any values of x that make the numerator and denominator equal to zero simultaneously. In this case, the numerator factors to (x - 3)(x + 1) and the denominator factors to (x + 2)(x + 1), giving us the factored form (x - 3)(x + 1)/(x + 2)(x + 1). Canceling out the common factors, we are left with (x - 3)/(x + 2).

Therefore, the function (x) has a hole at x = 3, as this value of x makes the numerator and denominator zero.