Simplify the rational expression and write the restriction for the simplified rational expression.
x^3 - 8
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x^2 + xy - 2x - 2y
x^3-8 = (x-2)(x^2 + 2x + 4)
x^2 + xy - 2x - 2y = (x-2)(x+y)
so, the fraction is
x^2+2x+4
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x+y
as long as x-2 and x+y are not zero
-6x+5=47 x
To simplify the rational expression (x^3 - 8) / (x^2 + xy - 2x - 2y), we can factor the numerator and the denominator.
Numerator: x^3 - 8
This is a difference of cubes, which can be factored as (x - 2)(x^2 + 2x + 4).
Denominator: x^2 + xy - 2x - 2y
We can factor this by grouping:
x^2 + xy - 2x - 2y
= (x^2 - 2x) + (xy - 2y)
= x(x - 2) + y(x - 2)
= (x + y)(x - 2)
Now we can simplify the rational expression by canceling out the common factors:
[(x - 2)(x^2 + 2x + 4)] / [(x + y)(x - 2)]
After canceling out the common factor of (x - 2), the simplified rational expression is:
(x^2 + 2x + 4) / (x + y)
To find the restriction for the simplified rational expression, we need to identify the values of x and y that would make the denominator (x + y) equal to zero. Since division by zero is undefined, we cannot have (x + y) equal to zero.
Therefore, the restriction for the simplified rational expression is:
x + y ≠ 0