simplify the rational expressions. State any excluded values. 1. 3x-6
2. x - 2 ______
______
x^2 + 3x--10 x--2
please walk me through this
To simplify rational expressions, we need to simplify both the numerator and denominator separately and then simplify the fraction if possible. Let's simplify the given expressions one by one.
1. Simplifying the rational expression (3x - 6):
In this expression, there is no denominator, so we only need to simplify the numerator.
The numerator, 3x - 6, does not have any common factors. Hence, it is already in its simplified form.
The excluded value for this expression occurs when the denominator is zero. Therefore, we need to find the value(s) of x that make the denominator equal to zero.
Setting the denominator x - 2 equal to zero and solving for x:
x - 2 = 0
x = 2
So, the excluded value for the expression 3x - 6 is x = 2.
2. Simplifying the rational expression (x - 2) / (x^2 + 3x - 10):
In this expression, both the numerator (x - 2) and the denominator (x^2 + 3x - 10) can be simplified.
The numerator is already in its simplified form, so we move on to the denominator.
The denominator, x^2 + 3x - 10, can be factored as follows:
x^2 + 3x - 10 = (x + 5)(x - 2)
Now, we can rewrite the rational expression as (x - 2) / [(x + 5)(x - 2)].
Next, we can cancel out the common factor of (x - 2) from the numerator and denominator:
(x - 2) / [(x + 5)(x - 2)] = 1 / (x + 5)
After canceling out the common factor, the simplified form of the expression is 1 / (x + 5).
Again, we need to find the excluded value for this expression. The excluded value occurs when the denominator of the simplified expression is zero. Therefore, we equate the denominator x + 5 to zero and solve for x:
x + 5 = 0
x = -5
So, the excluded value for the expression (x - 2) / (x^2 + 3x - 10) is x = -5.
In summary:
1. The simplified form of the expression 3x - 6 is 3x - 6. The excluded value is x = 2.
2. The simplified form of the expression (x - 2) / (x^2 + 3x - 10) is 1 / (x + 5). The excluded value is x = -5.