Question

determine if the expression $5r^4 - 4r^4b^5$ 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 represents a polynomial. the polynomial is a
and has a degree of

Explanation:

Step1: Recall Polynomial Definition

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non - negative integer exponents of variables. In the expression \(5r^{4}-4r^{4}b^{5}\), the variables are \(r\) and \(b\), and the exponents of \(r\) are \(4\) (in both terms) and the exponent of \(b\) is \(5\) (in the second term), all of which are non - negative integers. The operations used are subtraction and multiplication of variables and coefficients. So, the expression is a polynomial.

Step2: Determine the Type of Polynomial

The number of terms in the polynomial \(5r^{4}-4r^{4}b^{5}\) is 2. A polynomial with two terms is called a binomial.

Step3: Calculate the Degree of the Polynomial

To find the degree of a term in a polynomial with multiple variables, we add the exponents of the variables in that term.

• For the term \(5r^{4}\), the degree is \(4\) (since the exponent of \(r\) is \(4\) and there is no \(b\) variable, we can consider the exponent of \(b\) as \(0\), and \(4 + 0=4\)).
• For the term \(-4r^{4}b^{5}\), the degree is \(4 + 5=9\).

The degree of the polynomial is the highest degree among its terms. So, the degree of the polynomial \(5r^{4}-4r^{4}b^{5}\) is \(9\).

Answer:

The given expression represents a polynomial. The polynomial is a binomial and has a degree of \(9\).