Question

determine if the expression $-2s - 3s^5 - r^3s$ is a polynomial or not. if it is a polynomial, state the type and degree of the polynomial. answer attempt 1 out of 3 the given expression a polynomial.

Explanation:

Step1: Recall Polynomial Definition

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. In the expression \(-2s - 3s^{5}-r^{3}s\), the variables are \(s\) and \(r\), and the exponents of the variables are \(1\), \(5\), \(3\) and \(1\) (for \(r^{3}s=r^{3}s^{1}\)) which are non - negative integers. The operations involved are subtraction and multiplication.

Step2: Determine if it's a Polynomial

Since the expression \(-2s - 3s^{5}-r^{3}s\) satisfies the definition of a polynomial (variables with non - negative integer exponents, and operations of addition, subtraction, multiplication), it is a polynomial.

Step3: Determine the Type and Degree

• Type: A polynomial with more than one term is called a polynomial (specifically a trinomial since it has three terms: \(-2s\), \(- 3s^{5}\), and \(-r^{3}s\)).
• Degree: To find the degree of a polynomial with multiple variables, we find the sum of the exponents of the variables in each term and then take the maximum of those sums.
• For the term \(-2s\): The exponent of \(s\) is \(1\), and since there is no \(r\), the sum of exponents is \(1\).
• For the term \(-3s^{5}\): The exponent of \(s\) is \(5\), and there is no \(r\), so the sum of exponents is \(5\).
• For the term \(-r^{3}s\): The exponent of \(r\) is \(3\) and the exponent of \(s\) is \(1\), so the sum of exponents is \(3 + 1=4\).

The maximum of \(1\), \(5\), and \(4\) is \(5\), so the degree of the polynomial is \(5\).

Answer:

The given expression \(\boldsymbol{\text{is}}\) a polynomial. It is a trinomial (a type of polynomial) with degree \(5\).