Look at each pair of expressions below. Identify the expression that is not a  polynomial from each (a and b). Give reasons for your answer. 

A. 
1) 2x+1 
2.) x^-3+2x²-7
3.) 2√x
4.) 2x³-3x²^1/2+x-4
5.) (x+5) (9x+1)² (x-4) 

B.
1.) 2/x+1 
2.) x³+2x ²-7 
3.) x√2
4.) 2x³-3x²+ x-4
5.) (x+5) (9x+1)^4 / (x-4

To identify which expressions are not polynomials, we need to review the key characteristics of a polynomial. A polynomial in a variable \( x \) is an expression made up of terms that involve powers of \( x \) with non-negative integer exponents and real coefficients. Additionally, the variable \( x \) must not appear in denominators, under square roots, or with negative exponents.

### A. 
1.) \( 2x + 1 \)  
   - **Polynomial**: All terms have non-negative integer exponents (1 and 0 for \( x \)).

2.) \( x^{-3} + 2x^2 - 7 \)  
   - **Not a polynomial**: The term \( x^{-3} \) involves a negative exponent, which violates the rules for polynomials.

3.) \( 2\sqrt{x} \)  
   - **Not a polynomial**: The term \( \sqrt{x} \) is equivalent to \( x^{1/2} \), which is not a non-negative integer exponent.

4.) \( 2x^3 - 3x^{1/2} + x - 4 \)  
   - **Not a polynomial**: The term \( x^{1/2} \) (square root) involves a fractional exponent.

5.) \( (x+5)(9x+1)^2(x-4) \)  
   - **Polynomial**: All terms inside the factored form will result in non-negative integer exponents for \( x \) when expanded.

### B.
1.) \( \frac{2}{x} + 1 \)  
   - **Not a polynomial**: The term \( \frac{2}{x} \) involves \( x \) in the denominator, which is not allowed in a polynomial.

2.) \( x^3 + 2x^2 - 7 \)  
   - **Polynomial**: All terms have non-negative integer exponents (3, 2, and 0 for \( x \)).

3.) \( x\sqrt{2} \)  
   - **Polynomial**: Since \( \sqrt{2} \) is a constant and \( x \) has an exponent of 1, this is a polynomial.

4.) \( 2x^3 - 3x^2 + x - 4 \)  
   - **Polynomial**: All terms have non-negative integer exponents.

5.) \( \frac{(x+5)(9x+1)^4}{x-4} \)  
   - **Not a polynomial**: The term \( \frac{1}{x-4} \) introduces a denominator with a variable, which disqualifies it as a polynomial.

### Summary:
- In **A**, the expressions that are **not polynomials** are: 2, 3, and 4.
- In **B**, the expressions that are **not polynomials** are: 1 and 5.

Would you like to explore the concept of polynomials further or have any questions?

#### Here are some related questions:
1. What are the key differences between polynomials and rational expressions?
2. How can you determine the degree of a polynomial in factored form?
3. Why are negative exponents not allowed in polynomials?
4. Can a constant (like 5 or \( \sqrt{2} \)) be considered a polynomial?
5. How do you expand factored polynomial expressions like \( (x+5)(9x+1)^2(x-4) \)?

#### Tip:
Always remember that polynomials cannot have variables in the denominator, inside square roots, or raised to negative or fractional powers.

Look at each pair of expressions below. Identify the expression that is not a polynomial from each (A and B). Give reasons for your answer.

Learn how to identify non-polynomial expressions by analyzing algebraic expressions with terms that have negative exponents, square roots, or variables in the denominator. Suitable for Grades 9-12, this problem involves core polynomial concepts and their properties.

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