Which of the following expressions are polynomials ? In case of a polynomial , write its degree.
(i) `x^(5)-2x^(3)+x+sqrt(3)`
(ii) `y^(3)+sqrt(3)y`
(iii) `t^(2)-(2)/(5)t+sqrt(5)`
(iv) `x^(100)-1`
(v) `(1)/(sqrt(2))x^(2)-sqrt(2)x+2`
(vi) `x^(-2)+2x^(-1)+3`
(vii) 1
(viii) `(-3)/(5)`
(ix) `(x^(2))/(2)-(2)/(x^(2))`
(x) `root(3)(2)x^(2)-8`
(xi) `(1)/(2x^(2))`
(xii) `(1)/(sqrt(5))x^(1//2)+1`
(xiii) `(3)/(5)x^(2)-(7)/(3)x+9`
(xiv) `x^(4)-x^(3//2)+x-3`
(xv) `2x^(3)+3x^(2)+sqrt(x)-1`

Answer

To determine whether each of the given expressions is a polynomial, we need to check if all the powers of the variables in the expression are non-negative integers. If they are, we will also identify the degree of the polynomial, which is the highest power of the variable in the expression.

Let's analyze each expression step by step:

### (i) \( x^{5} - 2x^{3} + x + \sqrt{3} \)
- **Analysis**: The powers of \( x \) are 5, 3, and 1, which are all non-negative integers. The term \( \sqrt{3} \) is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 5.

### (ii) \( y^{3} + \sqrt{3}y \)
- **Analysis**: The powers of \( y \) are 3 and 1, which are both non-negative integers. The term \( \sqrt{3} \) is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 3.

### (iii) \( t^{2} - \frac{2}{5}t + \sqrt{5} \)
- **Analysis**: The powers of \( t \) are 2 and 1, which are non-negative integers. The term \( \sqrt{5} \) is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 2.

### (iv) \( x^{100} - 1 \)
- **Analysis**: The power of \( x \) is 100, which is a non-negative integer. The term -1 is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 100.

### (v) \( \frac{1}{\sqrt{2}}x^{2} - \sqrt{2}x + 2 \)
- **Analysis**: The powers of \( x \) are 2 and 1, which are non-negative integers. The term \( 2 \) is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 2.

### (vi) \( x^{-2} + 2x^{-1} + 3 \)
- **Analysis**: The powers of \( x \) are -2 and -1, which are negative integers. This does not satisfy the condition for being a polynomial.
- **Conclusion**: No, it is not a polynomial.

### (vii) \( 1 \)
- **Analysis**: This is a constant term.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The degree of a constant polynomial is 0.

### (viii) \( -\frac{3}{5} \)
- **Analysis**: This is also a constant term.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The degree of a constant polynomial is 0.

### (ix) \( \frac{x^{2}}{2} - \frac{2}{x^{2}} \)
- **Analysis**: The powers of \( x \) are 2 and -2. The term \( -\frac{2}{x^{2}} \) has a negative power.
- **Conclusion**: No, it is not a polynomial.

### (x) \( \sqrt[3]{2}x^{2} - 8 \)
- **Analysis**: The power of \( x \) is 2, which is a non-negative integer. The term -8 is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 2.

### (xi) \( \frac{1}{2x^{2}} \)
- **Analysis**: The power of \( x \) is -2, which is negative.
- **Conclusion**: No, it is not a polynomial.

### (xii) \( \frac{1}{\sqrt{5}}x^{\frac{1}{2}} + 1 \)
- **Analysis**: The power of \( x \) is \( \frac{1}{2} \), which is not a non-negative integer.
- **Conclusion**: No, it is not a polynomial.

### (xiii) \( \frac{3}{5}x^{2} - \frac{7}{3}x + 9 \)
- **Analysis**: The powers of \( x \) are 2 and 1, which are non-negative integers. The term 9 is a constant.
- **Conclusion**: Yes, it is a polynomial.
- **Degree**: The highest power is 2.

### (xiv) \( x^{4} - x^{\frac{3}{2}} + x - 3 \)
- **Analysis**: The power of \( x \) is \( \frac{3}{2} \), which is not a non-negative integer.
- **Conclusion**: No, it is not a polynomial.

### (xv) \( 2x^{3} + 3x^{2} + \sqrt{x} - 1 \)
- **Analysis**: The power of \( x \) is \( \frac{1}{2} \) in the term \( \sqrt{x} \), which is not a non-negative integer.
- **Conclusion**: No, it is not a polynomial.

### Summary of Results:
1. \( x^{5} - 2x^{3} + x + \sqrt{3} \) - Yes, Degree: 5
2. \( y^{3} + \sqrt{3}y \) - Yes, Degree: 3
3. \( t^{2} - \frac{2}{5}t + \sqrt{5} \) - Yes, Degree: 2
4. \( x^{100} - 1 \) - Yes, Degree: 100
5. \( \frac{1}{\sqrt{2}}x^{2} - \sqrt{2}x + 2 \) - Yes, Degree: 2
6. \( x^{-2} + 2x^{-1} + 3 \) - No
7. \( 1 \) - Yes, Degree: 0
8. \( -\frac{3}{5} \) - Yes, Degree: 0
9. \( \frac{x^{2}}{2} - \frac{2}{x^{2}} \) - No
10. \( \sqrt[3]{2}x^{2} - 8 \) - Yes, Degree: 2
11. \( \frac{1}{2x^{2}} \) - No
12. \( \frac{1}{\sqrt{5}}x^{\frac{1}{2}} + 1 \) - No
13. \( \frac{3}{5}x^{2} - \frac{7}{3}x + 9 \) - Yes, Degree: 2
14. \( x^{4} - x^{\frac{3}{2}} + x - 3 \) - No
15. \( 2x^{3} + 3x^{2} + \sqrt{x} - 1 \) - No