Which of the following expressions is not a polynomial ?

Options

A. `5a ^(2) b ^(2) + 6 ab - 2 ab ^(2)`
B. `- 2a ^(3) b ^(3) + 3 ab - 3 ab ^(2) c + 7 abc ^(2)`
C. `-13 x ^(2) + 14 xy - 15 y ^(2)`
D. `root 3 ( x ) + root4 ( y) `

Answer

To determine which of the following expressions is not a polynomial, we first need to understand the definition of a polynomial. A polynomial is an algebraic expression that consists of variables and coefficients, where the variables have non-negative integer powers (whole numbers).

Let's analyze each expression step by step:

1. **Expression 1**: \( x^2 + 3x + 2 \)
- The powers of \( x \) are 2 (from \( x^2 \)), 1 (from \( 3x \)), and 0 (from the constant term 2).
- All powers are non-negative integers.
- **Conclusion**: This is a polynomial.

2. **Expression 2**: \( 1 + x + x^3 \)
- The powers of \( x \) are 0 (from 1), 1 (from \( x \)), and 3 (from \( x^3 \)).
- All powers are non-negative integers.
- **Conclusion**: This is a polynomial.

3. **Expression 3**: \( 5x^4 - 2x^2 + 7 \)
- The powers of \( x \) are 4 (from \( 5x^4 \)), 2 (from \( -2x^2 \)), and 0 (from the constant term 7).
- All powers are non-negative integers.
- **Conclusion**: This is a polynomial.

4. **Expression 4**: \( x^{\frac{1}{3}} + y^{\frac{1}{4}} \)
- The powers of \( x \) and \( y \) are \( \frac{1}{3} \) and \( \frac{1}{4} \) respectively.
- Neither \( \frac{1}{3} \) nor \( \frac{1}{4} \) are non-negative integers.
- **Conclusion**: This is not a polynomial.

Based on the analysis, the expression that is not a polynomial is **Expression 4: \( x^{\frac{1}{3}} + y^{\frac{1}{4}} \)**.

### Summary of Steps:
1. Define what a polynomial is.
2. Analyze each expression for non-negative integer powers.
3. Identify which expression does not meet the polynomial criteria.