To determine which polynomial is written in standard form, we need to check that the exponents of [tex]\(x\)[/tex] in each polynomial are listed in descending order. Let's examine each polynomial one by one.
### Polynomial 1: [tex]\(2x - 8x^4 + 16x^5\)[/tex]
- The polynomial has terms with [tex]\(x\)[/tex] exponents of 1, 4, and 5.
- Ordering the terms by their exponents in descending order: [tex]\(16x^5 - 8x^4 + 2x\)[/tex].
- The given polynomial [tex]\(2x - 8x^4 + 16x^5\)[/tex] is not in this order. Hence, it is not in standard form.
### Polynomial 2: [tex]\(-16x^2 - 9x + 12\)[/tex]
- The polynomial has terms with [tex]\(x\)[/tex] exponents of 2, 1, and 0.
- Ordering the terms by their exponents in descending order: [tex]\(-16x^2 - 9x + 12\)[/tex].
- The given polynomial [tex]\(-16x^2 - 9x + 12\)[/tex] is already in this order. Hence, it is in standard form.
### Polynomial 3: [tex]\(15x - 6x^2 + 3\)[/tex]
- The polynomial has terms with [tex]\(x\)[/tex] exponents of 1, 2, and 0.
- Ordering the terms by their exponents in descending order: [tex]\(-6x^2 + 15x + 3\)[/tex].
- The given polynomial [tex]\(15x - 6x^2 + 3\)[/tex] is not in this order. Hence, it is not in standard form.
### Polynomial 4: [tex]\(-7x^7 + 9x^3 - 11x\)[/tex]
- The polynomial has terms with [tex]\(x\)[/tex] exponents of 7, 3, and 1.
- Ordering the terms by their exponents in descending order: [tex]\(-7x^7 + 9x^3 - 11x\)[/tex].
- The given polynomial [tex]\(-7x^7 + 9x^3 - 11x\)[/tex] is already in this order. Hence, it is in standard form.
So, the polynomials in standard form are:
1. [tex]\(-16x^2 - 9x + 12\)[/tex]
2. [tex]\(-7x^7 + 9x^3 - 11x\)[/tex]
This means the polynomials that are in standard form are Polynomials 2 and 4.
