To determine whether the expression $-x^4 y^2 \sqrt{2}$ is a polynomial, let's break it down:

1. Definition of a Polynomial: A polynomial is an algebraic expression composed of variables and constants, where the variables are raised to non-negative integer powers, and the coefficients are real numbers. Polynomials do not include radicals with variables, fractional powers, or negative exponents.

2. Analyzing the Expression:

   • The expression $-x^4 y^2 \sqrt{2}$ contains variables $x$ and $y$, raised to non-negative integer powers (4 and 2, respectively).
   • The constant $\sqrt{2}$ is a real number (approximately 1.414), not a variable.

Since the variables are raised to non-negative integer powers, and the coefficient $\sqrt{2}$ is a constant, the expression qualifies as a polynomial.

3. Type of Polynomial:

   • The expression has two variables, $x$ and $y$, so it is a multivariable polynomial.

4. Degree of the Polynomial:

   • The degree of a polynomial is the sum of the exponents of the variables in the term with the highest degree.
   • In this case, the highest degree term is $x^4 y^2$, and the total degree is $4 + 2 = 6$.

Thus, the given expression $-x^4 y^2 \sqrt{2}$ is a polynomial. It is a multivariable polynomial of degree 6.

Would you like more details or have any questions about this explanation?

Here are 5 related questions to explore:

1. What other types of expressions are not considered polynomials?
2. How do you classify polynomials based on the number of terms?
3. Can the degree of a polynomial be a fraction or negative number?
4. What is the difference between a univariate and a multivariate polynomial?
5. How do you determine the degree of a polynomial with multiple terms?

Tip: Always check the exponents of variables and the nature of coefficients when determining whether an expression is a polynomial.