find the domain of each rational expression in set notation.

x^2-36 divided by 3x

7w-2 divided by 16w^2-1

Then do both rational expressions have excluded values in their domains? If yes, explain why they are to be excluded from the domains. If no, explain why no exclusions are necessary.

domains of polynomials are all reals

for rational functions, exclude points where the denominator is zero. After all, division by zero is not defined.

6b-3/3b^2-12

To find the domain of a rational expression, we need to identify any values of the variables that would make the expression undefined. In a rational expression, the denominator cannot be equal to zero, as division by zero is undefined.

Let's determine the domain of each rational expression:

1. (x^2-36) / (3x):
To find the domain of this expression, we need to determine when the denominator, 3x, would be zero. Setting the denominator equal to zero, we have:
3x = 0
x = 0

Therefore, x cannot be zero since it would result in division by zero, which is undefined. Hence, the domain of this expression is all real numbers except x = 0. In set notation, we can write the domain as follows:
Domain: {x | x ∈ ℝ, x ≠ 0}

2. (7w-2) / (16w^2-1):
Similarly, we need to find when the denominator, 16w^2-1, would be zero. Setting the denominator equal to zero, we have:
16w^2 - 1 = 0

This equation can be rewritten as:
(4w - 1)(4w + 1) = 0

Solving for w, we find two possible values:
4w - 1 = 0 --> 4w = 1 --> w = 1/4
4w + 1 = 0 --> 4w = -1 --> w = -1/4

So, the denominator equals zero when w = 1/4 or w = -1/4. Thus, these values must be excluded from the domain. The domain of this expression is all real numbers except w = 1/4 and w = -1/4. In set notation, we can write the domain as follows:
Domain: {w | w ∈ ℝ, w ≠ 1/4, w ≠ -1/4}

Now, regarding the excluded values in the domains:

1. For the expression (x^2-36)/(3x), the excluded value is x = 0. This exclusion is necessary because dividing by zero is undefined in mathematics. If x were allowed to be zero, the expression would have no well-defined value.

2. Similarly, for the expression (7w-2)/(16w^2-1), the excluded values are w = 1/4 and w = -1/4. These exclusions are necessary because they would make the denominator equal to zero, resulting in division by zero, which is undefined.

Hence, both rational expressions have excluded values in their domains to ensure the expressions are mathematically valid and well-defined.