Which of the following are polynomials ?
`(i) x^(2)-3x+1 " " (ii) x^(2)+5x+2 " "(iii) x-(1)/(y) " " (iv) x^(7)+8 " " (v) x^(3)+sqrt(x)-2`
`(vi) sqrt(2)x^(2)+x-1 " " (vii) (3x-1)(x+5) " " (viii) (x-(3)/(x))(x+2) " " (ix)2x^(2)-1`
`(x) x+(1)/(sqrt(x))+2`

Answer

To determine which of the given expressions are polynomials, we need to check if each expression meets the criteria for being a polynomial. A polynomial must have non-negative integer powers of its variables.

Let's analyze each expression one by one:

1. **Expression (i):** \( x^2 - 3x + 1 \)
- All the powers of \( x \) are \( 2 \) and \( 1 \), which are non-negative integers.
- **Conclusion:** This is a polynomial (specifically, a quadratic polynomial).

2. **Expression (ii):** \( x^2 + 5x + 2 \)
- Again, all the powers of \( x \) are \( 2 \) and \( 1 \), which are non-negative integers.
- **Conclusion:** This is a polynomial (specifically, a quadratic polynomial).

3. **Expression (iii):** \( x - \frac{1}{y} \)
- The term \( \frac{1}{y} \) can be rewritten as \( y^{-1} \), which has a negative exponent.
- **Conclusion:** This is not a polynomial.

4. **Expression (iv):** \( x^7 + 8 \)
- The power of \( x \) is \( 7 \), which is a non-negative integer, and \( 8 \) is a constant.
- **Conclusion:** This is a polynomial (specifically, a polynomial of degree 7).

5. **Expression (v):** \( x^3 + \sqrt{x} - 2 \)
- The term \( \sqrt{x} \) can be rewritten as \( x^{1/2} \), which has a fractional exponent.
- **Conclusion:** This is not a polynomial.

6. **Expression (vi):** \( \sqrt{2}x^2 + x - 1 \)
- The coefficients are \( \sqrt{2} \) (which is a constant) and the powers of \( x \) are \( 2 \) and \( 1 \), which are non-negative integers.
- **Conclusion:** This is a polynomial (specifically, a quadratic polynomial).

7. **Expression (vii):** \( (3x - 1)(x + 5) \)
- We can expand this: \( 3x^2 + 15x - x - 5 = 3x^2 + 14x - 5 \).
- All the powers of \( x \) are \( 2 \) and \( 1 \), which are non-negative integers.
- **Conclusion:** This is a polynomial (specifically, a quadratic polynomial).

8. **Expression (viii):** \( (x - \frac{3}{x})(x + 2) \)
- The term \( \frac{3}{x} \) can be rewritten as \( 3x^{-1} \), which has a negative exponent.
- **Conclusion:** This is not a polynomial.

9. **Expression (ix):** \( 2x^2 - 1 \)
- The power of \( x \) is \( 2 \), which is a non-negative integer, and \( -1 \) is a constant.
- **Conclusion:** This is a polynomial (specifically, a quadratic polynomial).

10. **Expression (x):** \( x + \frac{1}{\sqrt{x}} + 2 \)
- The term \( \frac{1}{\sqrt{x}} \) can be rewritten as \( x^{-1/2} \), which has a negative exponent.
- **Conclusion:** This is not a polynomial.

### Summary of Results:
- Polynomials: (i), (ii), (iv), (vi), (vii), (ix)
- Not Polynomials: (iii), (v), (viii), (x)