Let's examine each expression to determine if it is a polynomial.

Part 1 of 4

$\frac{6}{7} x^4 y$ This expression is a polynomial. It is a monomial since it consists of a single term with variables having non-negative integer exponents.

Part 2 of 4

$4x + \frac{9}{x} - 5$ This expression is not a polynomial. The term $\frac{9}{x}$ can be rewritten as $9x^{-1}$, which has a negative exponent.

Part 3 of 4

$\sqrt{-9 + y^7 + 2y}$ This expression is not a polynomial. The square root operation applied to a polynomial results in an expression that is generally not a polynomial.

Part 4 of 4

$3 - 4x^8 + 5y$ This expression is a polynomial. It is a trinomial since it consists of three terms with variables having non-negative integer exponents.

To summarize:

1. $\frac{6}{7} x^4 y$ is a polynomial.
2. $4x + \frac{9}{x} - 5$ is not a polynomial.
3. $\sqrt{-9 + y^7 + 2y}$ is not a polynomial.
4. $3 - 4x^8 + 5y$ is a polynomial.

Would you like any more details or have any questions?

Here are five questions you might ask next:

1. How can I determine the degree of a polynomial?
2. What are the different types of polynomials?
3. Can you explain the properties of polynomial functions?
4. How do you perform polynomial long division?
5. What are polynomial identities?

Tip: A polynomial is defined as an expression that can have constants, variables, and exponents, which are non-negative integers, and it must not involve division by a variable.