Certainly! Let's analyze each algebraic expression to determine whether it is a polynomial.
1. Expression: [tex]\(2 x^3 - \frac{1}{x}\)[/tex]
- Analysis: A polynomial does not contain negative exponents or variables in the denominator. In this expression, [tex]\(\frac{1}{x}\)[/tex] introduces a term with a negative exponent ([tex]\(x^{-1}\)[/tex]).
- Conclusion: This expression is not a polynomial.
2. Expression: [tex]\(x^3 y - 3 x^2 + 6 x\)[/tex]
- Analysis: This expression has terms with positive integer exponents and involves only multiplication, subtraction, or addition of variables. It represents a valid polynomial.
- Conclusion: This expression is a polynomial.
3. Expression: [tex]\(y^2 + 5 y - \sqrt{3}\)[/tex]
- Analysis: This expression has terms with positive integer exponents and a constant term ([tex]\(- \sqrt{3}\)[/tex], which is allowed as a constant). It complies with polynomial rules.
- Conclusion: This expression is a polynomial.
4. Expression: [tex]\(2 - \sqrt{4 x}\)[/tex]
- Analysis: A polynomial does not include variables under a radical sign. The term [tex]\(\sqrt{4 x}\)[/tex] can be rewritten as [tex]\(2 \sqrt{x}\)[/tex], which is not a polynomial term because [tex]\(\sqrt{x}\)[/tex] is the same as [tex]\(x^{1/2}\)[/tex], a fractional exponent.
- Conclusion: This expression is not a polynomial.
5. Expression: [tex]\(-x + \sqrt{6}\)[/tex]
- Analysis: This expression consists of a linear term [tex]\(-x\)[/tex] and a constant term [tex]\(\sqrt{6}\)[/tex]. The square root of a constant is still a constant, which does not violate any polynomial rules.
- Conclusion: This expression is a polynomial.
6. Expression: [tex]\(-\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4}\)[/tex]
- Analysis: This expression consists of terms with positive integer exponents and constant coefficients. This aligns with the rules for polynomials.
- Conclusion: This expression is a polynomial.
### Summary:
The following expressions are polynomials:
- [tex]\(x^3 y - 3 x^2 + 6 x\)[/tex]
- [tex]\(y^2 + 5 y - \sqrt{3}\)[/tex]
- [tex]\(-x + \sqrt{6}\)[/tex]
- [tex]\(-\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4}\)[/tex]
Therefore, the polynomials are:
[tex]\[ \boxed{x^3 y - 3 x^2 + 6 x, y^2 + 5 y - \sqrt{3}, -x + \sqrt{6}, -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4}} \][/tex]
