Simplify each rational expression. State all restrictions on the variables
(𝑥^2−5𝑥−6)(𝑥^2−5𝑥+4) / (2𝑥−8)(2𝑥−2)
First, we can simplify each of the factors individually:
For the first factor, (𝑥^2−5𝑥−6), we can factor it as (𝑥−6)(𝑥+1).
For the second factor, (𝑥^2−5𝑥+4), we can factor it as (𝑥−4)(𝑥−1).
For the third factor, (2𝑥−8), we can factor out a 2, resulting in 2(𝑥−4).
For the fourth factor, (2𝑥−2), we can factor out a 2, resulting in 2(𝑥−1).
Now, we can write the simplified form of the rational expression:
((𝑥−6)(𝑥+1)(𝑥−4)(𝑥−1))/(2(𝑥−4)⋅ 2(𝑥−1))
Next, we can cancel out common factors in the numerator and denominator:
((𝑥−6)(𝑥+1))/(2⋅ 2)
Simplifying further, we have:
(𝑥−6)(𝑥+1)/4
The restrictions on the variables occur when a factor in the denominator is equal to 0. In this case, we need to ensure that (𝑥−4) and (𝑥−1) are not equal to 0. Therefore, the restrictions are:
𝑥 ≠ 4 and 𝑥 ≠ 1.