Answer :
Certainly! The question requires us to determine which of the given polynomials is in standard form. A polynomial is in standard form when its terms are ordered from the highest degree to the lowest degree.
Let's examine each polynomial one by one:
### Polynomial 1: [tex]\(2 x^4 + 6 + 24 x^5\)[/tex]
- The terms are [tex]\( 24 x^5 \)[/tex], [tex]\( 2 x^4 \)[/tex], and [tex]\( 6 \)[/tex].
- The degrees of these terms are [tex]\( 5 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 24 x^5 + 2 x^4 + 6 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 2: [tex]\(6 x^2 - 9 x^3 + 12 x^4\)[/tex]
- The terms are [tex]\( 12 x^4 \)[/tex], [tex]\( -9 x^3 \)[/tex], and [tex]\( 6 x^2 \)[/tex].
- The degrees of these terms are [tex]\( 4 \)[/tex], [tex]\( 3 \)[/tex], and [tex]\( 2 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 12 x^4 - 9 x^3 + 6 x^2 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 3: [tex]\(19 x + 6 x^2 + 2\)[/tex]
- The terms are [tex]\( 6 x^2 \)[/tex], [tex]\( 19 x \)[/tex], and [tex]\( 2 \)[/tex].
- The degrees of these terms are [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 6 x^2 + 19 x + 2 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 4: [tex]\(23 x^9 - 12 x^4 + 19\)[/tex]
- The terms are [tex]\( 23 x^9 \)[/tex], [tex]\( -12 x^4 \)[/tex], and [tex]\( 19 \)[/tex].
- The degrees of these terms are [tex]\( 9 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranged from highest degree to lowest, it is already written as [tex]\( 23 x^9 - 12 x^4 + 19 \)[/tex].
### Conclusion
Among the given polynomials, only the fourth polynomial, [tex]\(23 x^9 - 12 x^4 + 19\)[/tex], is already in standard form where the terms are ordered from the highest degree to the lowest degree.
Thus, the polynomial that is in standard form is:
[tex]\[ 23 x^9 - 12 x^4 + 19 \][/tex]
And it is the fourth polynomial.
Let's examine each polynomial one by one:
### Polynomial 1: [tex]\(2 x^4 + 6 + 24 x^5\)[/tex]
- The terms are [tex]\( 24 x^5 \)[/tex], [tex]\( 2 x^4 \)[/tex], and [tex]\( 6 \)[/tex].
- The degrees of these terms are [tex]\( 5 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 24 x^5 + 2 x^4 + 6 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 2: [tex]\(6 x^2 - 9 x^3 + 12 x^4\)[/tex]
- The terms are [tex]\( 12 x^4 \)[/tex], [tex]\( -9 x^3 \)[/tex], and [tex]\( 6 x^2 \)[/tex].
- The degrees of these terms are [tex]\( 4 \)[/tex], [tex]\( 3 \)[/tex], and [tex]\( 2 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 12 x^4 - 9 x^3 + 6 x^2 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 3: [tex]\(19 x + 6 x^2 + 2\)[/tex]
- The terms are [tex]\( 6 x^2 \)[/tex], [tex]\( 19 x \)[/tex], and [tex]\( 2 \)[/tex].
- The degrees of these terms are [tex]\( 2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranging from highest degree to lowest, we get [tex]\( 6 x^2 + 19 x + 2 \)[/tex].
- This polynomial is not in standard form.
### Polynomial 4: [tex]\(23 x^9 - 12 x^4 + 19\)[/tex]
- The terms are [tex]\( 23 x^9 \)[/tex], [tex]\( -12 x^4 \)[/tex], and [tex]\( 19 \)[/tex].
- The degrees of these terms are [tex]\( 9 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 0 \)[/tex].
- Arranged from highest degree to lowest, it is already written as [tex]\( 23 x^9 - 12 x^4 + 19 \)[/tex].
### Conclusion
Among the given polynomials, only the fourth polynomial, [tex]\(23 x^9 - 12 x^4 + 19\)[/tex], is already in standard form where the terms are ordered from the highest degree to the lowest degree.
Thus, the polynomial that is in standard form is:
[tex]\[ 23 x^9 - 12 x^4 + 19 \][/tex]
And it is the fourth polynomial.
