In This Discussion You Are Assigned Two Rational Expressions

In This Discussion You Are Assigned Two Rational Expressions To Work

In this discussion, you are assigned two rational expressions to work on. Remember to factor all polynomials completely. If you want to refresh your factoring skills, review sections 5.1 -- 5.6 in your e-book. Read the following instructions in order and view the MAT222 Week 1 Discussion Example. Please complete the following problems according to your assigned number. (Instructors will assign each student their number.) If your assigned number is 14: Your first rational expression is 64k²; your second rational expression is s² + 2s – 15 / s² – 36. Explain in your own words what the meaning of domain is. Also, explain why a denominator cannot be zero. Find the domain for each of your two rational expressions. Write the domain of each rational expression in set notation (as demonstrated in the example). Do both of your rational expressions have excluded value in their domains? If yes, explain why they are to be excluded from the domains. If no, explain why no exclusions are necessary. Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Domain, Excluded value, Set, Factor, Real numbers.

Paper For Above instruction

The concept of the domain in mathematics refers to the set of all possible input values (usually represented by variables) for which a function or rational expression is defined and produces a valid output. In the context of rational expressions, the domain comprises the set of all real numbers that do not cause the expression to be undefined. A denominator cannot be zero because division by zero is undefined in mathematics; it would lead to an invalid or infinite result, which makes the expression meaningless within real numbers.

Understanding the domain involves identifying values that would make the denominator zero—that is, the excluded values. For my assigned rational expressions, the first expression is 64k². Since it contains no denominator, its domain includes all real numbers—meaning, there are no excluded values in this case. Therefore, the domain of the first rational expression is {k | k ∈ R}.

The second rational expression is (s² + 2s – 15) / (s² – 36). Here, the denominator s² – 36 is critical because it can potentially be zero, leading to an undefined expression. To find the excluded values, we factor s² – 36 as a difference of squares: (s – 6)(s + 6). Setting this equal to zero gives s – 6 = 0 or s + 6 = 0, which implies s = 6 or s = -6. These are the excluded values for the domain because substituting these values makes the denominator zero, rendering the expression undefined.

Hence, the domain of the second expression excludes s = 6 and s = -6. In set notation, the domain is {s | s ∈ R, s ≠ 6, s ≠ -6}. These values are excluded because including them would violate the rule that the denominator cannot be zero in rational expressions. Since the expression's denominator involves a quadratic that factors into linear factors, the excluded values are the roots of the denominator, factoring into linear expressions that give zero. In this way, the domain reflects all real numbers except those that produce undefined results due to zero denominators.

In conclusion, the domain of rational expressions is a fundamental concept because it defines the scope within which the functions are valid. Excluded values are necessary to identify because they prevent division by zero. For expressions without a denominator, all real numbers are included in the set. For those with denominators, the set of allowable inputs excludes specific values that make the denominator zero, ensuring the expression remains valid within the domain.

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