Simplify each rational expression. State all restrictions on the variables.
𝑥^2(𝑥+2) / 𝑥^3 × (𝑥−4)(𝑥−1) / (𝑥+2)(𝑥−4)
To simplify the rational expression, we'll start by canceling out the common factors.
First, we can cancel out the common factor of (𝑥 + 2) in the numerator and denominator:
(x^2 * (x + 2)) / (x^3 * (x - 4)(x - 1)) * ((x - 4)(x - 1)) / ((x + 2)(x - 4))
Cancelling out (x + 2) from the numerator and denominator, we get:
(x^2 * 1) / (x^3 * (x - 4)(x - 1)) * (1) / ((1)(x - 4))
Continuing to simplify, we can cancel out (x - 4) from the numerator and denominator:
(x^2 * 1) / (x^3 * (x - 1)) * (1) / (1)
This further simplifies to:
x^2 / (x^3 * (x - 1))
Therefore, the simplified rational expression is x^2 / (x^3 * (x - 1)).
Restrictions on the variables:
We need to consider values of x that make the denominator equal to zero, as dividing by zero is undefined. In this case, the denominator is (x^3 * (x - 1)).
The restrictions on the variables are:
1) x ≠ 0 (because x^3 is in the denominator)
2) x ≠ 1 (because x - 1 is in the denominator)