The polynomial 18r²s + 6r + 7s² contains three terms, which can be classified based on their degree and structure:
- Term Analysis: The polynomial consists of the following terms:
- 18r²s: This is a term of degree 3, as the sum of the exponents (2 from r and 1 from s) equals 3.
- 6r: This term has a degree of 1, as the exponent of r is 1 and there are no other variables.
- 7s²: This term has a degree of 2, as the exponent of s is 2.
Overall Degree: The overall degree of the polynomial is determined by the term with the highest degree, which in this case is 18r²s with a degree of 3.
Coefficient Significance: The coefficients of the terms are:
- 18 for the term 18r²s
- 6 for the term 6r
- 7 for the term 7s²
All coefficients are positive, indicating that the polynomial’s value will increase as the values of r and s increase.
Variable Dependencies: The polynomial highlights a relationship among the variables r and s where the degree of each variable affects the overall structure and output of the polynomial. Increasing r will significantly influence the term with r², while increasing s will primarily affect the term with s². Thus, understanding how each variable contributes to the polynomial can help in applications such as optimization.
Summary: In summary, the polynomial 18r²s + 6r + 7s² is of degree 3, with diverse term degrees and positive coefficients, illustrating a multi-dimensional relationship between the variables r and s. This polynomial can be explored further for its roots, graphing behavior, or application in various mathematical contexts.