Let's analyze each of the algebraic expressions provided and give the best name for each:

1. $6x^7 + 18x^5 - 9x^2 + 17$:

   • The highest power of $x$ is 7 (degree of the polynomial), so it is a 7th-degree polynomial. It also contains multiple terms, making it a polynomial.

2. $8x^7 + 4$:

   • The highest power of $x$ is 7. It has two terms, so it is also a 7th-degree binomial (since it has two terms).

3. $2x^9$:

   • This is a single term with the highest power of $x$ being 9. It is a monomial and a 9th-degree monomial.

4. $20x^5 - 9x^2 + 16$:

   • The highest power of $x$ is 5. It contains three terms, so it is a 5th-degree trinomial.

5. $6x^4 + \frac{7}{x^3} - 18x^2 + 9$:

   • This expression contains terms with negative powers of $x$ (due to the $\frac{7}{x^3}$ term). Thus, it is not a standard polynomial but can be classified as a rational expression due to the division by $x^3$.

Would you like further details or have any questions?

Related Questions:

1. What makes a polynomial an nth-degree polynomial?
2. How do we differentiate between a binomial and a trinomial?
3. What is a rational expression in algebra?
4. How do terms with negative exponents affect the classification of expressions?
5. Can a monomial ever contain a fraction?

Tip: A polynomial's degree is determined by the highest exponent of the variable present in the expression.