Question

determine if the expression $5r^4 - 4r^4s^5$ 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 dropdown a polynomial.

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dropdown options: represents, does not represent

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 \(5r^{4}-4r^{4}s^{5}\), the variables are \(r\) and \(s\), the exponents of \(r\) are \(4\) (in both terms) and the exponent of \(s\) is \(5\), all of which are non - negative integers. The operations involved are subtraction and multiplication (of variables and coefficients).

Step2: Determine if it's a polynomial

Since the expression \(5r^{4}-4r^{4}s^{5}\) satisfies the definition of a polynomial (variables with non - negative integer exponents, and only addition, subtraction, and multiplication operations), it represents a polynomial.

Step3: Determine the type and degree (additional for completeness)

• Type: It has two terms (\(5r^{4}\) and \(- 4r^{4}s^{5}\)), so it is a binomial.
• Degree: To find the degree of a term in a polynomial with multiple variables, we add the exponents of the variables in the term. For the term \(5r^{4}\), the degree is \(4\) (since \(s^{0}\) is implicitly there, and \(4 + 0=4\)). For the term \(-4r^{4}s^{5}\), the degree is \(4 + 5=9\). The degree of the polynomial is the highest degree of its terms, so the degree is \(9\).

Answer:

The given expression \(\boldsymbol{\text{represents}}\) a polynomial. (It is a binomial polynomial with degree 9)