Final answer:
The simplified expression is x/(x + 7), after factoring and canceling out the common (x - 1) term in the numerator and denominator. Excluded values are x = 1 and x = -7 because they make the original denominator zero.
Step-by-step explanation:
To simplify the rational expression x² - x over (x - 1)(x + 7) we must first factor the numerator. We can factor out an x to get x(x - 1).
Now, the expression looks like this: x(x - 1) over (x - 1)(x + 7). We see that (x - 1) is a common factor in both the numerator and the denominator, so we can eliminate it.
Simplifying this we get x over (x + 7). However, we must also determine the excluded values. These are the values of x that would make the original denominator equal to zero, which are x = 1 and x = -7. Thus, the simplified expression is x/(x + 7) with excluded values of 1 and -7.
To check that this is reasonable, we can plug in a value for x (other than 1 and -7) into both the original and simplified expressions to ensure they produce the same result. Doing this with x = 2, for example, would give the same result for both, confirming the simplification is correct.