Simplify the rational expression. State any restrictions on the variable.
t^2+4t-12/t^2-4
Wow, oobleck, you are rude.
To simplify the rational expression (t^2+4t-12)/(t^2-4), we can factor both the numerator and denominator.
The numerator can be factored as (t+6)(t-2).
The denominator can be factored using the difference of squares formula as (t+2)(t-2).
Therefore, the simplified form of the rational expression is (t+6)(t-2)/(t+2)(t-2).
Now, in order to find any restrictions on the variable, we need to look at the denominator. Since we have (t-2) in the denominator, the expression is undefined when t = 2. So, the restriction on the variable is t ≠ 2.
To simplify the rational expression, we'll factor both the numerator and the denominator, then cancel out any common factors.
The numerator, t^2 + 4t - 12, can be factored as (t + 6)(t - 2).
The denominator, t^2 - 4, is a difference of squares, and can be factored as (t - 2)(t + 2).
Now, we can cancel out the common factor of (t - 2) from both the numerator and the denominator.
Therefore, the simplified rational expression is (t + 6)/(t + 2).
Restrictions on the variable:
To determine any restrictions on the variable, we need to look for values of t that would make the denominator equal to zero, as dividing by zero is undefined.
In this case, when t + 2 = 0, t = -2. Therefore, the value t = -2 is not allowed as it would result in division by zero.
Hence, the restriction on the variable is t ≠ -2.
assuming the usual sloppiness with parentheses,
(t^2+4t-12)/(t^2-4) = (t+6)(t-2) / (t+2)(t-2)
Now it's easy. Exclude any values for t where the denominator is zero, then cancel the (t-2) factor.