Simplify each rational expression. State all restrictions on the variables.
(𝑥^2−5𝑥−6)(𝑥^2−5𝑥+4) / (2𝑥−8)(2𝑥−2)
To simplify the expression, we need to factor both the numerator and denominator and then cancel out any common factors.
Let's begin:
The numerator can be factored as follows:
(𝑥^2−5𝑥−6)(𝑥^2−5𝑥+4) = (𝑥−6)(𝑥+1)(𝑥−4)(𝑥−1)
The denominator can be factored as follows:
(2𝑥−8)(2𝑥−2) = 2(𝑥−4)2(𝑥−1) = 4(𝑥−4)(𝑥−1)
Now, we can rewrite the rational expression as:
(𝑥−6)(𝑥+1)(𝑥−4)(𝑥−1) / 4(𝑥−4)(𝑥−1)
Next, we can cancel out the common factors:
(𝑥−6)(𝑥+1) / 4
Therefore, the simplified rational expression is:
(𝑥−6)(𝑥+1) / 4
The restriction on the variable is that x cannot be equal to 4 or 1, since these values would make the denominator zero.