Question

determine if the expression $-s^5 + 8s^5t^2$ is a polynomial or not. if it is a polynomial state the type and degree of the polynomial. monomial binomial trinomial none of the above represents a polynomial. the polynomial is a and has a degree of .

Explanation:

Step1: Check if it's a polynomial

A polynomial is an expression with non - negative integer exponents on variables and no division by variables. In the expression \(-s^{5}+8s^{5}t^{2}\), the exponents of \(s\) and \(t\) are non - negative integers (\(5\) for \(s\) in the first term, \(5\) for \(s\) and \(2\) for \(t\) in the second term) and there is no division by variables. So it is a polynomial.

Step2: Determine the type of polynomial

A monomial has 1 term, a binomial has 2 terms, and a trinomial has 3 terms. The expression \(-s^{5}+8s^{5}t^{2}\) has two terms: \(-s^{5}\) and \(8s^{5}t^{2}\). So it is a binomial.

Step3: Find the degree of the polynomial

The degree of a term in a polynomial with multiple variables is the sum of the exponents of the variables in that term. For the term \(-s^{5}\), the degree is \(5\) (since the exponent of \(s\) is \(5\) and there is no \(t\) term, so we consider the exponent of \(t\) as \(0\), and \(5 + 0=5\)). For the term \(8s^{5}t^{2}\), the degree is \(5 + 2=7\). The degree of the polynomial is the highest degree of its terms. So the degree is \(7\).

Answer:

The expression \(-s^{5}+8s^{5}t^{2}\) represents a polynomial. The polynomial is a binomial and has a degree of \(7\).