Ansswer: No, in general, a rational expression and its simplified forms are equivalent, due domain restrictions.
Explanation:
1) A rational expression is the quotient of two poynomials. For example,
R(x) = p(x) / q(x)
2) To simplify such kind of expressions, you need to find the common factors of the numerator, p(x), and the denominator, q(x), and then cancell them out.
3) By cancelling some factors in the denominator, you are implicitly changing the domain of the entire expression.
That is because you cannot cancell out expressions in the denominator that are equal to zero, since the division by zero is not defined.
The only valid form to cancell those factors in the denominator is that you state that the values of x that make the denominator zero are excluded.
2) See this example
a) Let it be p(x) = x²- 1, and q(x) = x - 1
i) then R(x) = [x² - 1] / (x - 1)
ii) factor x² - 1: (x + 1)(x - 1)
iii) rewrite the rational expression: (x + 1) (x - 1) / (x - 1)
iv) cancell out the common factors: x + 1
Then, as result, the simplified expression of R(x) = [x² - 1] / (x - 1) is x + 1.
But they are not equivalent, because R(x) is not defined for x = 1, whlie x + 1 is defined in all the Real numbers.
