Answer :
We need to identify which polynomial is written with its terms in descending order of the exponents. A polynomial is in standard form when its terms are arranged from the highest power of [tex]$x$[/tex] to the lowest.
Let’s analyze each option:
1. Option 1:
[tex]$$2x^4 + 6 + 24x^5$$[/tex]
The degrees of the terms are:
- [tex]$2x^4$[/tex] has degree [tex]$4$[/tex],
- [tex]$6$[/tex] (a constant) has degree [tex]$0$[/tex],
- [tex]$24x^5$[/tex] has degree [tex]$5$[/tex].
Since the order given is degree [tex]$4$[/tex], then [tex]$0$[/tex], then [tex]$5$[/tex], it is not in descending order.
2. Option 2:
[tex]$$6x^2 - 9x^3 + 12x^4$$[/tex]
The degrees of the terms are:
- [tex]$6x^2$[/tex] has degree [tex]$2$[/tex],
- [tex]$-9x^3$[/tex] has degree [tex]$3$[/tex],
- [tex]$12x^4$[/tex] has degree [tex]$4$[/tex].
The order here is [tex]$2$[/tex], [tex]$3$[/tex], [tex]$4$[/tex], which is ascending rather than descending.
3. Option 3:
[tex]$$19x + 6x^2 + 2$$[/tex]
The degrees of the terms are:
- [tex]$19x$[/tex] has degree [tex]$1$[/tex],
- [tex]$6x^2$[/tex] has degree [tex]$2$[/tex],
- [tex]$2$[/tex] (a constant) has degree [tex]$0$[/tex].
This order is [tex]$1$[/tex], [tex]$2$[/tex], [tex]$0$[/tex], which is not in descending order (the correct descending order would be [tex]$2$[/tex], [tex]$1$[/tex], [tex]$0$[/tex]).
4. Option 4:
[tex]$$23x^9 - 12x^4 + 19$$[/tex]
The degrees of the terms are:
- [tex]$23x^9$[/tex] has degree [tex]$9$[/tex],
- [tex]$-12x^4$[/tex] has degree [tex]$4$[/tex],
- [tex]$19$[/tex] (a constant) has degree [tex]$0$[/tex].
The degrees are in descending order: [tex]$9$[/tex], [tex]$4$[/tex], [tex]$0$[/tex].
Therefore, the polynomial in standard form is:
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
Let’s analyze each option:
1. Option 1:
[tex]$$2x^4 + 6 + 24x^5$$[/tex]
The degrees of the terms are:
- [tex]$2x^4$[/tex] has degree [tex]$4$[/tex],
- [tex]$6$[/tex] (a constant) has degree [tex]$0$[/tex],
- [tex]$24x^5$[/tex] has degree [tex]$5$[/tex].
Since the order given is degree [tex]$4$[/tex], then [tex]$0$[/tex], then [tex]$5$[/tex], it is not in descending order.
2. Option 2:
[tex]$$6x^2 - 9x^3 + 12x^4$$[/tex]
The degrees of the terms are:
- [tex]$6x^2$[/tex] has degree [tex]$2$[/tex],
- [tex]$-9x^3$[/tex] has degree [tex]$3$[/tex],
- [tex]$12x^4$[/tex] has degree [tex]$4$[/tex].
The order here is [tex]$2$[/tex], [tex]$3$[/tex], [tex]$4$[/tex], which is ascending rather than descending.
3. Option 3:
[tex]$$19x + 6x^2 + 2$$[/tex]
The degrees of the terms are:
- [tex]$19x$[/tex] has degree [tex]$1$[/tex],
- [tex]$6x^2$[/tex] has degree [tex]$2$[/tex],
- [tex]$2$[/tex] (a constant) has degree [tex]$0$[/tex].
This order is [tex]$1$[/tex], [tex]$2$[/tex], [tex]$0$[/tex], which is not in descending order (the correct descending order would be [tex]$2$[/tex], [tex]$1$[/tex], [tex]$0$[/tex]).
4. Option 4:
[tex]$$23x^9 - 12x^4 + 19$$[/tex]
The degrees of the terms are:
- [tex]$23x^9$[/tex] has degree [tex]$9$[/tex],
- [tex]$-12x^4$[/tex] has degree [tex]$4$[/tex],
- [tex]$19$[/tex] (a constant) has degree [tex]$0$[/tex].
The degrees are in descending order: [tex]$9$[/tex], [tex]$4$[/tex], [tex]$0$[/tex].
Therefore, the polynomial in standard form is:
[tex]$$23x^9 - 12x^4 + 19.$$[/tex]
