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

Explanation:

Step1: Recall polynomial definition

A polynomial in one - variable $x$ is an expression of the form $a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0$, where $n$ is a non - negative integer and $a_i$ are constants.

Step2: Analyze each option

• For $4x^2-3x+\frac{2}{x}=4x^2 - 3x+2x^{-1}$, the term $2x^{-1}$ has a negative exponent, so it is not a polynomial.
• For $-6x^3 + x^2-\sqrt{5}$, it is of the form $a_3x^3+a_2x^2 + a_0$ where $a_3=-6,a_2 = 1,a_0=-\sqrt{5}$, and the exponents of $x$ are non - negative integers. So it is a polynomial.
• For $8x^2+\sqrt{x}=8x^2+x^{\frac{1}{2}}$, the term $x^{\frac{1}{2}}$ has a non - integer exponent, so it is not a polynomial.
• For $-2x^4+\frac{3}{2x}=-2x^4+\frac{3}{2}x^{-1}$, the term $\frac{3}{2}x^{-1}$ has a negative exponent, so it is not a polynomial.

Answer:

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