Question

which algebraic expression is a polynomial?
\\( 3m^2n - \frac{2m}{n} + \frac{1}{n} \\)
\\( \frac{2mn}{5} - \frac{\sqrt{m}}{4} + 4m^5 \\)
\\( \frac{4m^3}{n^2} - 3mn^5 + \sqrt{8} \\)
\\( 7mn + \frac{3m}{2} + \frac{5n}{4} \\)

Explanation:

Step1: Recall Polynomial Definition

A polynomial is an expression with non - negative integer exponents on variables, no variable in denominator, no square - root (or other radical) of a variable.

Step2: Analyze Option 1

Expression: \(3m^{2}n-\frac{2m}{n}+\frac{1}{n}\). There are terms \(\frac{2m}{n}\) and \(\frac{1}{n}\) where \(n\) is in the denominator (equivalent to \(n^{- 1}\), exponent is - 1, not a non - negative integer). So not a polynomial.

Step3: Analyze Option 2

Expression: \(\frac{2mn}{5}-\frac{\sqrt{m}}{4}+4m^{5}\). The term \(\frac{\sqrt{m}}{4}= \frac{m^{\frac{1}{2}}}{4}\), exponent of \(m\) is \(\frac{1}{2}\), not an integer. So not a polynomial.

Step4: Analyze Option 3

Expression: \(\frac{4m^{3}}{n^{2}}-3mn^{5}+\sqrt{8}\). The term \(\frac{4m^{3}}{n^{2}} = 4m^{3}n^{-2}\), exponent of \(n\) is - 2, not a non - negative integer. So not a polynomial.

Step5: Analyze Option 4

Expression: \(7mn+\frac{3m}{2}+\frac{5n}{4}\). All exponents on variables \(m\) and \(n\) are non - negative integers (exponent of \(m\) in \(7mn\) is 1, in \(\frac{3m}{2}\) is 1; exponent of \(n\) in \(7mn\) is 1, in \(\frac{5n}{4}\) is 1), and no variables in denominator or under radical. So it is a polynomial.

Answer:

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