Final answer:
The zeros of the function (X^2-2X-8)/(X^2-1) are found by setting the numerator equal to zero and factoring. The zeros are X = 4 and X = -2; X = 1 and X = -1 are not zeros but points of discontinuity.
Step-by-step explanation:
To find the zeros of the function (X^2-2X-8)/(X^2-1), we need to focus on the numerator, because zeros occur when the numerator equals zero. The denominator informs us about potential restrictions or asymptotes, which are values that X cannot be because they would make the function undefined. Here's how we find the zeros step by step:
- Set the numerator equal to zero: X^2 - 2X - 8 = 0.
- Factor the quadratic equation: (X - 4)(X + 2) = 0.
- Find the zeros from the factored form by setting each factor equal to zero: X - 4 = 0 or X + 2 = 0, so the zeros are X = 4 and X = -2.
Please note that X cannot be 1 or -1 since these values would make the denominator equal to zero, which is not allowed in a fraction. Hence, even though X = 1 and X = -1 are solutions to the denominator when set to zero, they are not zeros of the entire function, but rather points of discontinuity.