which algebraic expression is a polynomial?\n$3m^{2}n-\frac{2m}{n}+\frac{1}{n}$\n$\frac{2mn}{5}-\frac{sqrt{m}…

Explanation:

Step1: Recall polynomial definition

A polynomial in two variables (m) and (n) consists of terms where the exponents of the variables are non - negative integers and there are no variables in the denominator or under a radical.

Step2: Analyze the first option

The expression (3m^{2}n-\frac{2m}{n}+\frac{1}{n}=3m^{2}n - 2mn^{-1}+n^{-1}) has variables (n) in the denominator, so it is not a polynomial.

Step3: Analyze the second option

The expression (\frac{2mn}{5}-\frac{\sqrt{m}}{4}+4m^{5}=\frac{2}{5}mn-\frac{1}{4}m^{\frac{1}{2}}+4m^{5}) has a variable (m) under a square - root (i.e., the exponent of (m) in the second term is (\frac{1}{2})), so it is not a polynomial.

Step4: Analyze the third option

The expression (\frac{4m^{3}}{n^{2}}-3mn^{5}+\sqrt{8}=4m^{3}n^{-2}-3mn^{5}+\sqrt{8}) has a variable (n) in the denominator, so it is not a polynomial.

Step5: Analyze the fourth option

The expression (7mn+\frac{3m}{2}+\frac{5n}{4}=\frac{7}{1}mn+\frac{3}{2}m+\frac{5}{4}n) has non - negative integer exponents for the variables (m) and (n) and no variables in the denominator or under a radical. So it is a polynomial.

Answer:

(7mn+\frac{3m}{2}+\frac{5n}{4})