Final answer:
The factored form of the rational expression a²-b²/b+a is (a+b)(a-b)/(b+a), which reduces to a-b after eliminating the term a+b. The excluded value is -b, since we cannot divide by zero. The reduction is reasonable, and it's checked with a numerical example.
Step-by-step explanation:
To reduce the rational expression a²-b²/b+a to the lowest terms and identify the excluded values of the variables, let's start by factoring the numerator which is a difference of squares:
a²-b² can be factored as (a+b)(a-b).
The factored form of the rational expression is (a+b)(a-b) / (b+a).
We notice that a+b and b+a are the same (since addition is commutative), so we can eliminate terms wherever possible:
(a+b)(a-b) / (b+a) = a-b.
So the reduced form is a-b.
To find the excluded values of the variable, we look at the original denominator b+a. It can't be zero because division by zero is undefined. Therefore, the excluded value is -b. Listing it from least to greatest gives us just the single value: -b.
Finally, let's check the answer to see if it is reasonable. Considering an example with numerical values for a and b (excluding the excluded values), for instance, let's say a = 2, b = 3, substituting these into the reduced form a-b gives us 2 - 3 = -1, which matches what we would get if we applied the same substitution into the original expression. Therefore, it is reasonable.