To determine which polynomial is in standard form, we need to write each polynomial with its terms in descending order of their degree (the highest power of \( x \) to the lowest). Let's examine each polynomial step by step.
1. \( 9 + 2x - 8x^4 + 16x^5 \)
- The terms are: \( 9 \) (degree 0), \( 2x \) (degree 1), \( -8x^4 \) (degree 4), \( 16x^5 \) (degree 5).
- Ordered by degree (highest to lowest): \( 16x^5 - 8x^4 + 2x + 9 \).
2. \( 12x^5 - 6x^2 - 9x + 12 \)
- The terms are: \( 12x^5 \) (degree 5), \( -6x^2 \) (degree 2), \( -9x \) (degree 1), \( 12 \) (degree 0).
- Ordered by degree (highest to lowest): \( 12x^5 - 6x^2 - 9x + 12 \).
3. \( 13x^5 + 11x - 6x^2 + 5 \)
- The terms are: \( 13x^5 \) (degree 5), \( 11x \) (degree 1), \( -6x^2 \) (degree 2), \( 5 \) (degree 0).
- Ordered by degree (highest to lowest): \( 13x^5 - 6x^2 + 11x + 5 \).
4. \( 7x^7 + 14x^9 - 17x + 25 \)
- The terms are: \( 7x^7 \) (degree 7), \( 14x^9 \) (degree 9), \( -17x \) (degree 1), \( 25 \) (degree 0).
- Ordered by degree (highest to lowest): \( 14x^9 + 7x^7 - 17x + 25 \).
To be in standard form, the polynomials must have their exponents arranged in descending order from left to right.
Examining each polynomial:
- \( 16x^5 - 8x^4 + 2x + 9 \): This polynomial is in standard form.
- \( 12x^5 - 6x^2 - 9x + 12 \): This polynomial is in standard form.
- \( 13x^5 - 6x^2 + 11x + 5 \): This polynomial is in standard form.
- \( 14x^9 + 7x^7 - 17x + 25 \): This polynomial is in standard form.
All these polynomials can be converted to a standard form with rearranging, but the original question seems to be asking which one is already written in the standard format.
Given the original polynomials, the polynomial that is closest to standard form without rearrangement is option 2:
[tex]\[
12x^5 - 6x^2 - 9x + 12
\][/tex]
Thus, the answer is:
[tex]\[
12x^5 - 6x^2 - 9x + 12
\][/tex]
It already lists terms in decreasing order of exponents.
