To determine which polynomial is in standard form, we need to check if the terms of each polynomial are written in descending order of their exponents.
Let's examine each polynomial step-by-step:
1. Polynomial: [tex]\(2x^4 + 6 + 24x^5\)[/tex]
- Terms: [tex]\(2x^4\)[/tex], [tex]\(6\)[/tex] (constant), [tex]\(24x^5\)[/tex]
- Exponents: 4, 0 (for the constant term), 5
- The exponents should be in descending order: 5, 4, 0
Given the terms [tex]\(2x^4\)[/tex], [tex]\(24x^5\)[/tex], and [tex]\(6\)[/tex], they are not in descending order as [tex]\(4\)[/tex] is less than [tex]\(5\)[/tex].
2. Polynomial: [tex]\(6x^2 - 9x^3 + 12x^4\)[/tex]
- Terms: [tex]\(6x^2\)[/tex], [tex]\(-9x^3\)[/tex], [tex]\(12x^4\)[/tex]
- Exponents: 2, 3, 4
- The exponents should be in descending order: 4, 3, 2
Given the terms [tex]\(6x^2\)[/tex], [tex]\(-9x^3\)[/tex], and [tex]\(12x^4\)[/tex], they are not in descending order as [tex]\(2\)[/tex] is less than [tex]\(3\)[/tex] and [tex]\(4\)[/tex].
3. Polynomial: [tex]\(19x + 6x^2 + 2\)[/tex]
- Terms: [tex]\(19x\)[/tex], [tex]\(6x^2\)[/tex], [tex]\(2\)[/tex] (constant)
- Exponents: 1, 2, 0 (for the constant term)
- The exponents should be in descending order: 2, 1, 0
Given the terms [tex]\(19x\)[/tex], [tex]\(6x^2\)[/tex], and [tex]\(2\)[/tex], they are not in descending order as [tex]\(1\)[/tex] is less than [tex]\(2\)[/tex].
4. Polynomial: [tex]\(23x^9 - 12x^4 + 19\)[/tex]
- Terms: [tex]\(23x^9\)[/tex], [tex]\(-12x^4\)[/tex], [tex]\(19\)[/tex] (constant)
- Exponents: 9, 4, 0 (for the constant term)
- The exponents should be in descending order: 9, 4, 0
Given the terms [tex]\(23x^9\)[/tex], [tex]\(-12x^4\)[/tex], and [tex]\(19\)[/tex], they are in descending order as [tex]\(9\)[/tex] is greater than [tex]\(4\)[/tex] and both are greater than [tex]\(0\)[/tex].
Therefore, the polynomial that is in standard form is:
[tex]\[ 23x^9 - 12x^4 + 19 \][/tex]
This corresponds to the fourth polynomial in the given list.
