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 − 2 s − 3 s 5 − r 3 s is analyzed term by term to check if it meets the criteria of a polynomial.
The expression involves two variables, r and s , and all exponents are non-negative integers.
The degree of each term is determined, and the highest degree is identified as the degree of the polynomial.
The given expression is a polynomial in two variables of degree 5, so the answer is that the expression is a polynomial.

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.

Checking for Polynomial Criteria 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. Let's examine the given expression term by term:



The first term is − 2 s . The variable is s and its exponent is 1, which is a non-negative integer.
The second term is − 3 s 5 . The variable is s and its exponent is 5, which is a non-negative integer.
The third term is − r 3 s . The variables are r and s . The exponent of r is 3 and the exponent of s is 1, both of which are non-negative integers.

Since all the exponents are non-negative integers, the given expression is a polynomial.

Determining the Type of Polynomial Now, let's determine the type of the polynomial. Since the expression involves two variables, r and s , it is a polynomial in two variables.

Finding the Degree of the Polynomial Next, we need to find the degree of the polynomial. The degree of a polynomial is the highest degree of its terms. The degree of a term is the sum of the exponents of the variables in that term.



The degree of the term − 2 s is 1.
The degree of the term − 3 s 5 is 5.
The degree of the term − r 3 s is 3 + 1 = 4 .

The highest degree among these terms is 5. Therefore, the degree of the polynomial is 5.

Conclusion Therefore, the given expression is a polynomial in two variables of degree 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 polynomials is crucial for solving real-world problems in these fields.

Answered by GinnyAnswer | 2025-07-03

The expression − 2 s − 3 s 5 − r 3 s is a polynomial because all its terms meet the criteria for polynomials. It is a polynomial in two variables, specifically r and s , with a degree of 5. Therefore, we conclude that it is indeed a polynomial of degree 5.
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Answered by Anonymous | 2025-07-04