To determine which polynomial is in standard form, we need to ensure that the polynomial is written in descending order of the power of [tex]\( x \)[/tex]. This means the term with the highest power of [tex]\( x \)[/tex] should be written first, followed by the term with the next highest power, and so on, until the constant term (if any) appears last.
Let's analyze each polynomial step-by-step.
1. Polynomial: [tex]\( 9 + 2x - 8x^4 + 16x^5 \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( 0, 1, 4, 5 \)[/tex]
- Order: [tex]\( 16x^5 - 8x^4 + 2x + 9 \)[/tex] (standard form would be descending)
- Analysis: This polynomial is not in descending order when initially given.
2. Polynomial: [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( 5, 2, 1, 0 \)[/tex]
- Order: [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex]
- Analysis: The terms are arranged in descending order of the powers of [tex]\( x \)[/tex]. This polynomial is in standard form.
3. Polynomial: [tex]\( 13x^5 + 11x - 6x^2 + 5 \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( 5, 1, 2, 0 \)[/tex]
- Order: [tex]\( 13x^5 - 6x^2 + 11x + 5 \)[/tex] (standard form would be [tex]\( x^5, x^2, x^1, x^0 \)[/tex])
- Analysis: This polynomial is not in descending order when initially given.
4. Polynomial: [tex]\( 7x^7 + 14x^9 - 17x + 25 \)[/tex]
- Powers of [tex]\( x \)[/tex]: [tex]\( 7, 9, 1, 0 \)[/tex]
- Order: [tex]\( 14x^9 + 7x^7 - 17x + 25 \)[/tex] (standard form would be [tex]\( x^9, x^7, x^1, x^0 \)[/tex])
- Analysis: This polynomial is not in descending order when initially given.
From the analysis, we see that the second polynomial, [tex]\( 12x^5 - 6x^2 - 9x + 12 \)[/tex], is the only one that is already in standard form.
Therefore, the polynomial in standard form is:
[tex]\[ 12x^5 - 6x^2 - 9x + 12 \][/tex]
Hence, the answer is:
[tex]\[ \boxed{12x^5 - 6x^2 - 9x + 12} \][/tex]
And within the given options, its index is:
[tex]\[ \boxed{2} \][/tex]
This corresponds to option (b) in your list.
