Determine if the expression $-2 s-3 s^5-r^3 s$ is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial.

The expression is a polynomial because all exponents are non-negative integers.
The polynomial has two variables: r and s .
The degree of each term is: 1, 5, and 4.
The degree of the polynomial is the highest degree among its terms, which is 5 ​ .

Explanation

Analyzing the Expression We are given the expression − 2 s − 3 s 5 − r 3 s and we need to determine if it is a polynomial. If it is, we need to state its type and degree.

Definition of a Polynomial A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Checking the Exponents In the given expression, the terms are − 2 s , − 3 s 5 , and − r 3 s . The exponents of the variables are 1, 5, 3, and 1, which are all non-negative integers. Therefore, the expression is a polynomial.

Identifying the Variables The variables in the polynomial are r and s . Since there are two variables, the polynomial is a polynomial in two variables.

Determining the Degree of Each Term Now, let's find the degree of each term:



The degree of the term − 2 s is 1 (since the exponent of s is 1).
The degree of the term − 3 s 5 is 5 (since the exponent of s is 5).
The degree of the term − r 3 s is 3 + 1 = 4 (since the exponents of r and s are 3 and 1, respectively).


Determining the Degree of the Polynomial The degree of the polynomial is the highest degree of any term in the polynomial. In this case, the degrees of the terms are 1, 5, and 4. The highest degree is 5. Therefore, the degree of the polynomial is 5.

Final Answer In conclusion, the expression − 2 s − 3 s 5 − r 3 s is a polynomial in two variables, r and s , and its degree is 5.


Examples
Polynomials are used in various fields such as physics, engineering, computer science, and economics. For example, in physics, polynomials can be used to model the trajectory of a projectile. In economics, polynomials can be used to model cost and revenue functions. Understanding the degree and type of a polynomial helps in analyzing and interpreting the behavior of the modeled phenomenon.

Answered by GinnyAnswer | 2025-07-03

The expression − 2 s − 3 s 5 − r 3 s is a polynomial because all of its exponents are non-negative integers. It has two variables, r and s , and the highest degree among its terms is 5. Therefore, the degree of the polynomial is 5.
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Answered by Anonymous | 2025-07-04