Question

which algebraic expression is a polynomial?
○ $4x^2 - 3x + \frac{2}{x}$
○ $-6x^3 + x^2 - \sqrt{5}$
○ $8x^2 + \sqrt{x}$
○ $-2x^4 + \frac{3}{2x}$

Explanation:

A polynomial is an expression where variables have non-negative integer exponents, no variables in denominators, and no square roots of variables.

1. $4x^2 - 3x + \frac{2}{x}$ has $\frac{2}{x}=2x^{-1}$, a negative exponent, so not a polynomial.
2. $-6x^3 + x^2 - \sqrt{5}$ has non-negative integer exponents for $x$, and $\sqrt{5}$ is a constant term, so this is a polynomial.
3. $8x^2 + \sqrt{x}$ has $\sqrt{x}=x^{\frac{1}{2}}$, a non-integer exponent, so not a polynomial.
4. $-2x^4 + \frac{3}{2x}$ has $\frac{3}{2x}=\frac{3}{2}x^{-1}$, a negative exponent, so not a polynomial.

Answer:

$\boldsymbol{-6x^3 + x^2 - \sqrt{5}}$