Find any excluded values of each rational expression.
1. t - 1/ t^2 - t
A: Excluded value: 1.
Simplify each rational expression, if possible. Identify any excluded values.
2. 3x^2/6x^3
A: 2x; x ≠ 2
3. 2n/n^2 - 3n
A: ?
4. s + 1/s^2 - 4s - 5
A: ?
5. 12 - 3x/x^2 - 8x + 16
A: ?
#1 also t=0, since t^2-t = t(t-1)
The original expression must be examined for excluded values.
(t-1)/(t^2-t) = 1/t, so t≠0. But dividing top and bottom by t-1 to get there means that t≠1, because you cannot divide by zero.
#2 huh? There is no x-2 in the denominator.
1/(2x) x≠0
#3
2n / n(n-3) = 2/(n-3), x≠0,3
Same as #1 for logic.
#4
(s+1) / (s+1)(s-5)
s≠-1,5
#5
3(4-x) / (x-4)^2 = -3/(x-4), x≠4
To find the excluded values, we need to identify any values of the variable that would result in division by zero.
1. For the rational expression (t - 1)/(t^2 - t), there is an excluded value of 1. This is because when t = 1, the denominator becomes 1^2 - 1 = 0, which is undefined.
2. For the rational expression (3x^2)/(6x^3), we can simplify to get (x^2)/(2x^3), and then further simplify to get 1/(2x). There is an excluded value of x = 0, because division by zero is undefined.
3. For the rational expression 2n/(n^2 - 3n), we need to factor the denominator. Factoring n out gives us n(n - 3). The excluded values are the solutions to the equation n(n - 3) = 0, which are n = 0 and n = 3. These values make the denominator zero, so they are excluded.
4. For the rational expression (s + 1)/(s^2 - 4s - 5), we need to factor the denominator. Factoring gives us (s - 5)(s + 1). The excluded values are the solutions to the equation (s - 5)(s + 1) = 0, which are s = 5 and s = -1. These values make the denominator zero, so they are excluded.
5. For the rational expression (12 - 3x)/(x^2 - 8x + 16), we need to factor the denominator. The denominator factors to (x - 4)(x - 4) or (x - 4)^2. There are no excluded values because (x - 4)^2 is never zero.
In summary:
1. Excluded value: 1
2. Excluded value: x ≠ 0
3. Excluded values: 0, 3
4. Excluded values: 5, -1
5. No excluded values.