To determine which of the given algebraic expressions is a polynomial, we need to understand the definition of a polynomial.
A polynomial in a single variable [tex]\( x \)[/tex] is an algebraic expression that consists of terms in the form [tex]\( a_n x^n \)[/tex], where [tex]\( a_n \)[/tex] is a coefficient and [tex]\( n \)[/tex] is a non-negative integer. Polynomials can be added, subtracted, and multiplied together, and they can also be scaled by multiplying or dividing by constants.
Given this, let's analyze each expression one by one:
1. [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
- [tex]\( 4x^2 \)[/tex] is a polynomial term.
- [tex]\( -3x \)[/tex] is a polynomial term.
- [tex]\( \frac{2}{x} = 2x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex] is not a polynomial.
2. [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex]
- [tex]\( -6x^3 \)[/tex] is a polynomial term.
- [tex]\( x^2 \)[/tex] is a polynomial term.
- [tex]\( -\sqrt{5} \)[/tex] is a constant term (coefficients can be constants or roots), which means it is part of a polynomial.
Hence, [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex] is a polynomial.
3. [tex]\( 8x^2 + \sqrt{x} \)[/tex]
- [tex]\( 8x^2 \)[/tex] is a polynomial term.
- [tex]\( \sqrt{x} = x^{1/2} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\( \frac{1}{2} \)[/tex], which is not an integer.
Hence, [tex]\( 8x^2 + \sqrt{x} \)[/tex] is not a polynomial.
4. [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
- [tex]\( -2x^4 \)[/tex] is a polynomial term.
- [tex]\( \frac{3}{2x} = \frac{3}{2} x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( -2x^4 + \frac{3}{2x} \)[/tex] is not a polynomial.
In conclusion, out of the given algebraic expressions, the only one that is a polynomial is:
[tex]\[ -6x^3 + x^2 - \sqrt{5} \][/tex]
