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.