Identify if the given is polynomial or not. If not, explain why. If it is a polynomial, classify according to number of terms.

Steps to Solve

1. Examine the first expression: $-7x + y$

This expression has two terms, $-7x$ and $y$. Both terms are coefficients with non-negative integer powers of the variables. Thus, it is a polynomial with 2 terms.

2. Examine the second expression: $\frac{4x}{y}$

This expression can be rewritten as $4xy^{-1}$. The term $y^{-1}$ has a negative exponent, which means it does not satisfy polynomial criteria. Therefore, this expression is not a polynomial.

3. Examine the third expression: $8mn^{-1}$

This expression can be rewritten as $8m n^{-1}$. Similar to the previous case, $n^{-1}$ has a negative exponent. As a result, this expression is also not a polynomial.

4. Examine the fourth expression: $x^2 - 5xy + 7y^2$

This expression contains three terms: $x^2$, $-5xy$, and $7y^2$. All terms have non-negative integer powers. Therefore, this is a polynomial with 3 terms.

5. Examine the fifth expression: $\sqrt{m} - n^4$

This can be rewritten as $m^{1/2} - n^4$. The term $m^{1/2}$ has a fractional exponent, which disqualifies it from being a polynomial. Thus, this expression is not a polynomial.

6. Examine the sixth expression: $\sqrt{6}uv^3$

This can be rewritten as $6^{1/2} u v^3$. Here, the term $6^{1/2}$ is a constant, and $u$ and $v^3$ represent variables with non-negative integer powers. Hence, it is a polynomial with 1 term.