Find the product in the simplest form. State any restrictions on the variable. Show all your work. X^2-9/x-2 • x^2-6x+8/ x^2-x-12
To find the product of the given expressions, we multiply the numerators and denominators separately and simplify if possible.
Numerator:
(x^2-9) * (x^2-6x+8)
= (x+3)(x-3) * (x-2)(x-4)
= (x+3)(x-3)(x-2)(x-4)
= (x-3)(x-2)(x+3)(x-4) (Rearranging the terms)
Denominator:
(x-2) * (x^2-x-12)
= (x-2) * (x-4)(x+3)
= (x-2)(x-4)(x+3)
Therefore, the product in the simplest form is:
[(x-3)(x-2)(x+3)(x-4)] / [(x-2)(x-4)(x+3)]
Now, we can simplify the expression. We notice that the terms (x-2)(x-4)(x+3) appear in both the numerator and denominator, so they cancel out.
The simplified expression is:
(x-3) / 1 (since (x-2)(x-4)(x+3) / (x-2)(x-4)(x+3) = 1)
Therefore, the final simplified expression is:
x-3
There are no restrictions on the variable x in this case.