To determine which polynomial is in standard form, we need to consider the term with the highest degree in each polynomial. The polynomial with the highest degree term written first is considered to be in standard form. Let's analyze each polynomial step-by-step:
1. [tex]\(9 + 2x - 8x^4 + 16x^5\)[/tex]
- The terms in this polynomial are: [tex]\( 9, 2x, -8x^4, 16x^5 \)[/tex]
- The highest degree term is [tex]\( 16x^5 \)[/tex]
2. [tex]\(12x^5 - 6x^2 - 9x + 12\)[/tex]
- The terms in this polynomial are: [tex]\( 12x^5, -6x^2, -9x, 12 \)[/tex]
- The highest degree term is [tex]\( 12x^5 \)[/tex]
3. [tex]\(13x^5 + 11x - 6x^2 + 5\)[/tex]
- The terms in this polynomial are: [tex]\( 13x^5, 11x, -6x^2, 5 \)[/tex]
- The highest degree term is [tex]\( 13x^5 \)[/tex]
4. [tex]\(7x^7 + 14x^9 - 17x + 25\)[/tex]
- The terms in this polynomial are: [tex]\( 7x^7, 14x^9, -17x, 25 \)[/tex]
- The highest degree term is [tex]\( 14x^9 \)[/tex]
Now, let's compare the highest degree terms of each polynomial:
- Polynomial 1: [tex]\( 16x^5 \)[/tex] (degree 5)
- Polynomial 2: [tex]\( 12x^5 \)[/tex] (degree 5)
- Polynomial 3: [tex]\( 13x^5 \)[/tex] (degree 5)
- Polynomial 4: [tex]\( 14x^9 \)[/tex] (degree 9)
The polynomial with the highest degree term, therefore, is Polynomial 4 with the term [tex]\( 14x^9 \)[/tex].
Polynomials in standard form are written with their terms in descending order of degree. Given that Polynomial 4 has its highest degree term [tex]\( 14x^9 \)[/tex] first, it is already in standard form.
So, the polynomial in standard form is:
[tex]\[ 7x^7 + 14x^9 - 17x + 25 \][/tex]
Thus, the correct answer is Polynomial 4.
