To determine which algebraic expression is a polynomial, we'll need to examine each expression individually. A polynomial is an expression that consists of variables and coefficients, composed using only addition, subtraction, multiplication, and non-negative integer exponents.
1. Expression: \( 4x^2 - 3x + \frac{2}{x} \)
- The first term \( 4x^2 \) is a polynomial term.
- The second term \( -3x \) is also a polynomial term.
- The third term \( \frac{2}{x} \) can be rewritten as \( 2x^{-1} \), which includes a negative exponent and thus disqualifies the expression as a polynomial.
- Conclusion: This expression is not a polynomial.
2. Expression: \( -6x^3 + x^2 - \sqrt{5} \)
- The first term \( -6x^3 \) is a polynomial term.
- The second term \( x^2 \) is also a polynomial term.
- The third term \( -\sqrt{5} \) is a constant, which is a polynomial term (since constants are polynomials with degree 0).
- Conclusion: This expression is a polynomial.
3. Expression: \( 8x^2 + \sqrt{x} \)
- The first term \( 8x^2 \) is a polynomial term.
- The second term \( \sqrt{x} \) can be rewritten as \( x^{1/2} \), which includes a fractional exponent and thus disqualifies the expression as a polynomial.
- Conclusion: This expression is not a polynomial.
4. Expression: \( -2x^4 + \frac{3}{2x} \)
- The first term \( -2x^4 \) is a polynomial term.
- The second term \( \frac{3}{2x} \) can be rewritten as \( \frac{3}{2} x^{-1} \), which includes a negative exponent and thus disqualifies the expression as a polynomial.
- Conclusion: This expression is not a polynomial.
Among the given options, the only expression that qualifies as a polynomial is:
[tex]\[ -6x^3 + x^2 - \sqrt{5} \][/tex]
Therefore, the answer is that there is one algebraic expression in the list that is a polynomial.
