Which of the following expressions are polynomials in one variable and which are not ? State reasons for your answer .
(i) `4x^(2)-3x+7`
(ii) `y^(2)+sqrt(2)`
(iii) `3sqrt(t)+tsqrt(2)`
(iv) `y=(2)/(y)`
(v) `x^(10)+y^(3)+t^(50)`

Answer

To determine which of the given expressions are polynomials in one variable and which are not, we will analyze each expression based on the definition of a polynomial. A polynomial in one variable is an expression that consists of terms with non-negative integer exponents of that variable and has no other variables involved.

Let's go through each expression step by step:

### Step 1: Analyze the first expression `4x^2 - 3x + 7`
- **Variables**: The only variable present is `x`.
- **Exponents**: The exponents of `x` are 2 (in `4x^2`), 1 (in `-3x`), and 0 (in the constant term `7`).
- **Conclusion**: All exponents are non-negative integers, and there is only one variable.
- **Result**: This is a polynomial in one variable.

### Step 2: Analyze the second expression `y^2 + √2`
- **Variables**: The only variable present is `y`.
- **Exponents**: The exponent of `y` is 2 (in `y^2`), and the term `√2` can be considered as a constant (which has an exponent of 0).
- **Conclusion**: All exponents are non-negative integers, and there is only one variable.
- **Result**: This is a polynomial in one variable.

### Step 3: Analyze the third expression `3√t + t√2`
- **Variables**: The variable present is `t`.
- **Exponents**: The term `3√t` can be rewritten as `3t^(1/2)`, which has an exponent of 1/2 (not a non-negative integer). The term `t√2` can be rewritten as `t^(1)t^(1/2)` which has an exponent of 1.
- **Conclusion**: Since one of the terms has a non-integer exponent (1/2), this does not satisfy the conditions for a polynomial.
- **Result**: This is not a polynomial.

### Step 4: Analyze the fourth expression `y = 2/y`
- **Variables**: The variable present is `y`.
- **Exponents**: Rearranging gives `y^2 = 2`, which implies `y` has an exponent of -1 when rearranged.
- **Conclusion**: The expression involves division by a variable, which leads to a negative exponent, violating the polynomial condition.
- **Result**: This is not a polynomial.

### Step 5: Analyze the fifth expression `x^10 + y^3 + t^50`
- **Variables**: The variables present are `x`, `y`, and `t`.
- **Exponents**: The exponents are 10 (in `x^10`), 3 (in `y^3`), and 50 (in `t^50`), all of which are non-negative integers.
- **Conclusion**: Although all exponents are non-negative integers, there are multiple variables.
- **Result**: This is not a polynomial in one variable.

### Final Summary:
- (i) `4x^2 - 3x + 7`: **Polynomial in one variable**
- (ii) `y^2 + √2`: **Polynomial in one variable**
- (iii) `3√t + t√2`: **Not a polynomial**
- (iv) `y = 2/y`: **Not a polynomial**
- (v) `x^10 + y^3 + t^50`: **Not a polynomial in one variable**